Find The First Partial Derivatives Of The Function.${ Z = (6x + 5y)^8 }$ { \frac{\partial Z}{\partial X} = \square \} ${ \frac{\partial Z}{\partial Y} = \square }$

by ADMIN 167 views

Introduction

In calculus, partial derivatives are a fundamental concept used to study the behavior of functions with multiple variables. The partial derivative of a function with respect to one of its variables is the derivative of the function with respect to that variable, while keeping the other variables constant. In this article, we will focus on finding the first partial derivatives of the function z=(6x+5y)8z = (6x + 5y)^8.

What are Partial Derivatives?

Partial derivatives are a way to measure the rate of change of a function with respect to one of its variables, while keeping the other variables constant. They are denoted by the symbol ∂z∂x\frac{\partial z}{\partial x} or ∂z∂y\frac{\partial z}{\partial y}, where zz is the function and xx and yy are the variables.

The Chain Rule

To find the partial derivatives of the function z=(6x+5y)8z = (6x + 5y)^8, we will use the chain rule. The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. In this case, the function zz is a composite function of the function f(u)=u8f(u) = u^8, where u=6x+5yu = 6x + 5y.

Finding the Partial Derivative with Respect to x

To find the partial derivative of the function zz with respect to xx, we will use the chain rule. The chain rule states that if we have a composite function z=f(u)z = f(u), where u=g(x)u = g(x), then the partial derivative of zz with respect to xx is given by:

∂z∂x=∂f∂u⋅∂u∂x\frac{\partial z}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}

In this case, we have z=(6x+5y)8z = (6x + 5y)^8, so we can write:

∂z∂x=∂f∂u⋅∂u∂x\frac{\partial z}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}

where f(u)=u8f(u) = u^8 and u=6x+5yu = 6x + 5y.

Calculating the Partial Derivative

To calculate the partial derivative, we need to find the derivatives of f(u)f(u) and uu with respect to xx. We have:

∂f∂u=8u7\frac{\partial f}{\partial u} = 8u^7

and

∂u∂x=6\frac{\partial u}{\partial x} = 6

Substituting these values into the chain rule formula, we get:

∂z∂x=8u7⋅6\frac{\partial z}{\partial x} = 8u^7 \cdot 6

Now, we need to substitute the value of uu into the equation. We have u=6x+5yu = 6x + 5y, so we can substitute this value into the equation:

∂z∂x=8(6x+5y)7⋅6\frac{\partial z}{\partial x} = 8(6x + 5y)^7 \cdot 6

Simplifying the equation, we get:

∂z∂x=48(6x+5y)7\frac{\partial z}{\partial x} = 48(6x + 5y)^7

Finding the Partial Derivative with Respect to y

To find the partial derivative of the function zz with respect to yy, we will use the chain rule. The chain rule states that if we have a composite function z=f(u)z = f(u), where u=g(y)u = g(y), then the partial derivative of zz with respect to yy is given by:

∂z∂y=∂f∂u⋅∂u∂y\frac{\partial z}{\partial y} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y}

In this case, we have z=(6x+5y)8z = (6x + 5y)^8, so we can write:

∂z∂y=∂f∂u⋅∂u∂y\frac{\partial z}{\partial y} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y}

where f(u)=u8f(u) = u^8 and u=6x+5yu = 6x + 5y.

Calculating the Partial Derivative

To calculate the partial derivative, we need to find the derivatives of f(u)f(u) and uu with respect to yy. We have:

∂f∂u=8u7\frac{\partial f}{\partial u} = 8u^7

and

∂u∂y=5\frac{\partial u}{\partial y} = 5

Substituting these values into the chain rule formula, we get:

∂z∂y=8u7⋅5\frac{\partial z}{\partial y} = 8u^7 \cdot 5

Now, we need to substitute the value of uu into the equation. We have u=6x+5yu = 6x + 5y, so we can substitute this value into the equation:

∂z∂y=8(6x+5y)7⋅5\frac{\partial z}{\partial y} = 8(6x + 5y)^7 \cdot 5

Simplifying the equation, we get:

∂z∂y=40(6x+5y)7\frac{\partial z}{\partial y} = 40(6x + 5y)^7

Conclusion

In this article, we have found the first partial derivatives of the function z=(6x+5y)8z = (6x + 5y)^8 with respect to xx and yy. We have used the chain rule to calculate the partial derivatives, and we have simplified the equations to get the final answers.

Partial Derivatives of a Function: A Comprehensive Guide

Introduction

In calculus, partial derivatives are a fundamental concept used to study the behavior of functions with multiple variables. The partial derivative of a function with respect to one of its variables is the derivative of the function with respect to that variable, while keeping the other variables constant. In this article, we will focus on finding the first partial derivatives of the function z=(6x+5y)8z = (6x + 5y)^8.

What are Partial Derivatives?

Partial derivatives are a way to measure the rate of change of a function with respect to one of its variables, while keeping the other variables constant. They are denoted by the symbol ∂z∂x\frac{\partial z}{\partial x} or ∂z∂y\frac{\partial z}{\partial y}, where zz is the function and xx and yy are the variables.

The Chain Rule

To find the partial derivatives of the function z=(6x+5y)8z = (6x + 5y)^8, we will use the chain rule. The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. In this case, the function zz is a composite function of the function f(u)=u8f(u) = u^8, where u=6x+5yu = 6x + 5y.

Finding the Partial Derivative with Respect to x

To find the partial derivative of the function zz with respect to xx, we will use the chain rule. The chain rule states that if we have a composite function z=f(u)z = f(u), where u=g(x)u = g(x), then the partial derivative of zz with respect to xx is given by:

∂z∂x=∂f∂u⋅∂u∂x\frac{\partial z}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}

In this case, we have z=(6x+5y)8z = (6x + 5y)^8, so we can write:

∂z∂x=∂f∂u⋅∂u∂x\frac{\partial z}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}

where f(u)=u8f(u) = u^8 and u=6x+5yu = 6x + 5y.

Calculating the Partial Derivative

To calculate the partial derivative, we need to find the derivatives of f(u)f(u) and uu with respect to xx. We have:

∂f∂u=8u7\frac{\partial f}{\partial u} = 8u^7

and

∂u∂x=6\frac{\partial u}{\partial x} = 6

Substituting these values into the chain rule formula, we get:

∂z∂x=8u7⋅6\frac{\partial z}{\partial x} = 8u^7 \cdot 6

Now, we need to substitute the value of uu into the equation. We have u=6x+5yu = 6x + 5y, so we can substitute this value into the equation:

∂z∂x=8(6x+5y)7⋅6\frac{\partial z}{\partial x} = 8(6x + 5y)^7 \cdot 6

Simplifying the equation, we get:

∂z∂x=48(6x+5y)7\frac{\partial z}{\partial x} = 48(6x + 5y)^7

Finding the Partial Derivative with Respect to y

To find the partial derivative of the function zz with respect to yy, we will use the chain rule. The chain rule states that if we have a composite function z=f(u)z = f(u), where u=g(y)u = g(y), then the partial derivative of zz with respect to yy is given by:

∂z∂y=∂f∂u⋅∂u∂y\frac{\partial z}{\partial y} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y}

In this case, we have z=(6x+5y)8z = (6x + 5y)^8, so we can write:

Q&A: Partial Derivatives of a Function

Q: What are partial derivatives?

A: Partial derivatives are a way to measure the rate of change of a function with respect to one of its variables, while keeping the other variables constant. They are denoted by the symbol ∂z∂x\frac{\partial z}{\partial x} or ∂z∂y\frac{\partial z}{\partial y}, where zz is the function and xx and yy are the variables.

Q: How do I find the partial derivative of a function?

A: To find the partial derivative of a function, you need to use the chain rule. The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. In this case, the function zz is a composite function of the function f(u)=u8f(u) = u^8, where u=6x+5yu = 6x + 5y.

Q: What is the chain rule?

A: The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. In this case, the function zz is a composite function of the function f(u)=u8f(u) = u^8, where u=6x+5yu = 6x + 5y. The chain rule states that if we have a composite function z=f(u)z = f(u), where u=g(x)u = g(x), then the partial derivative of zz with respect to xx is given by:

∂z∂x=∂f∂u⋅∂u∂x</span></p><h3><strong>Q:HowdoIcalculatethepartialderivativeusingthechainrule?</strong></h3><p>A:Tocalculatethepartialderivativeusingthechainrule,youneedtofindthederivativesof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>u</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">f(u)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">u</span><spanclass="mclose">)</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotationencoding="application/x−tex">u</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">u</span></span></span></span>withrespectto<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotationencoding="application/x−tex">x</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span></span></span></span>.Wehave:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac><mo>=</mo><mn>8</mn><msup><mi>u</mi><mn>7</mn></msup></mrow><annotationencoding="application/x−tex">∂f∂u=8u7</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">u</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8641em;"></span><spanclass="mord">8</span><spanclass="mord"><spanclass="mordmathnormal">u</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">7</span></span></span></span></span></span></span></span></span></span></span></span></p><p>and</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>6</mn></mrow><annotationencoding="application/x−tex">∂u∂x=6</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">x</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">u</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">6</span></span></span></span></span></p><p>Substitutingthesevaluesintothechainruleformula,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>8</mn><msup><mi>u</mi><mn>7</mn></msup><mo>⋅</mo><mn>6</mn></mrow><annotationencoding="application/x−tex">∂z∂x=8u7⋅6</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">x</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8641em;"></span><spanclass="mord">8</span><spanclass="mord"><spanclass="mordmathnormal">u</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">7</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">6</span></span></span></span></span></p><p>Now,weneedtosubstitutethevalueof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotationencoding="application/x−tex">u</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">u</span></span></span></span>intotheequation.Wehave<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow><annotationencoding="application/x−tex">u=6x+5y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">u</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8389em;vertical−align:−0.1944em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>,sowecansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>8</mn><mostretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mostretchy="false">)</mo><mn>7</mn></msup><mo>⋅</mo><mn>6</mn></mrow><annotationencoding="application/x−tex">∂z∂x=8(6x+5y)7⋅6</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">x</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">8</span><spanclass="mopen">(</span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">7</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">6</span></span></span></span></span></p><p>Simplifyingtheequation,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>48</mn><mostretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mostretchy="false">)</mo><mn>7</mn></msup></mrow><annotationencoding="application/x−tex">∂z∂x=48(6x+5y)7</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">x</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">48</span><spanclass="mopen">(</span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">7</span></span></span></span></span></span></span></span></span></span></span></span></p><h3><strong>Q:Whatisthepartialderivativeofthefunction<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mostretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mostretchy="false">)</mo><mn>8</mn></msup></mrow><annotationencoding="application/x−tex">z=(6x+5y)8</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0641em;vertical−align:−0.25em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">8</span></span></span></span></span></span></span></span></span></span></span>withrespectto<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>?</strong></h3><p>A:Tofindthepartialderivativeofthefunction<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mostretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mostretchy="false">)</mo><mn>8</mn></msup></mrow><annotationencoding="application/x−tex">z=(6x+5y)8</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0641em;vertical−align:−0.25em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">8</span></span></span></span></span></span></span></span></span></span></span>withrespectto<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>,wewillusethechainrule.Thechainrulestatesthatifwehaveacompositefunction<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>f</mi><mostretchy="false">(</mo><mi>u</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">z=f(u)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">u</span><spanclass="mclose">)</span></span></span></span>,where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mi>g</mi><mostretchy="false">(</mo><mi>y</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">u=g(y)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">u</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose">)</span></span></span></span>,thenthepartialderivativeof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span>withrespectto<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>isgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mimathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mimathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mrow><annotationencoding="application/x−tex">∂z∂y=∂f∂u⋅∂u∂y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">u</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">u</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><p>Inthiscase,wehave<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mostretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mostretchy="false">)</mo><mn>8</mn></msup></mrow><annotationencoding="application/x−tex">z=(6x+5y)8</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0641em;vertical−align:−0.25em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">8</span></span></span></span></span></span></span></span></span></span></span>,sowecanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mimathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mimathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mrow><annotationencoding="application/x−tex">∂z∂y=∂f∂u⋅∂u∂y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">u</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">u</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><p>where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>u</mi><mostretchy="false">)</mo><mo>=</mo><msup><mi>u</mi><mn>8</mn></msup></mrow><annotationencoding="application/x−tex">f(u)=u8</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">u</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8141em;"></span><spanclass="mord"><spanclass="mordmathnormal">u</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">8</span></span></span></span></span></span></span></span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow><annotationencoding="application/x−tex">u=6x+5y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">u</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8389em;vertical−align:−0.1944em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>.</p><h3><strong>Q:HowdoIcalculatethepartialderivativewithrespectto<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>?</strong></h3><p>A:Tocalculatethepartialderivativewithrespectto<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>,weneedtofindthederivativesof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>u</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">f(u)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">u</span><spanclass="mclose">)</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotationencoding="application/x−tex">u</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">u</span></span></span></span>withrespectto<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>.Wehave:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac><mo>=</mo><mn>8</mn><msup><mi>u</mi><mn>7</mn></msup></mrow><annotationencoding="application/x−tex">∂f∂u=8u7</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">u</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8641em;"></span><spanclass="mord">8</span><spanclass="mord"><spanclass="mordmathnormal">u</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">7</span></span></span></span></span></span></span></span></span></span></span></span></p><p>and</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>5</mn></mrow><annotationencoding="application/x−tex">∂u∂y=5</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal">u</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">5</span></span></span></span></span></p><p>Substitutingthesevaluesintothechainruleformula,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>8</mn><msup><mi>u</mi><mn>7</mn></msup><mo>⋅</mo><mn>5</mn></mrow><annotationencoding="application/x−tex">∂z∂y=8u7⋅5</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8641em;"></span><spanclass="mord">8</span><spanclass="mord"><spanclass="mordmathnormal">u</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">7</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">5</span></span></span></span></span></p><p>Now,weneedtosubstitutethevalueof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotationencoding="application/x−tex">u</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">u</span></span></span></span>intotheequation.Wehave<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow><annotationencoding="application/x−tex">u=6x+5y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">u</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8389em;vertical−align:−0.1944em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span></span>,sowecansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>8</mn><mostretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mostretchy="false">)</mo><mn>7</mn></msup><mo>⋅</mo><mn>5</mn></mrow><annotationencoding="application/x−tex">∂z∂y=8(6x+5y)7⋅5</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">8</span><spanclass="mopen">(</span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">7</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">5</span></span></span></span></span></p><p>Simplifyingtheequation,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mfrac><mrow><mimathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mimathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>40</mn><mostretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mostretchy="false">)</mo><mn>7</mn></msup></mrow><annotationencoding="application/x−tex">∂z∂y=40(6x+5y)7</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"style="margin−right:0.05556em;">∂</span><spanclass="mordmathnormal"style="margin−right:0.04398em;">z</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">40</span><spanclass="mopen">(</span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">7</span></span></span></span></span></span></span></span></span></span></span></span></p><h3><strong>Q:Whataresomecommonapplicationsofpartialderivatives?</strong></h3><p>A:Partialderivativeshavemanyapplicationsinvariousfields,includingphysics,engineering,economics,andcomputerscience.Somecommonapplicationsofpartialderivativesinclude:</p><ul><li>Findingthemaximumorminimumofafunction</li><li>Determiningtherateofchangeofafunction</li><li>Solvingoptimizationproblems</li><li>Modelingreal−worldphenomena</li></ul><h3><strong>Q:HowdoIusepartialderivativesinreal−worldapplications?</strong></h3><p>A:Tousepartialderivativesinreal−worldapplications,youneedtounderstandtheconceptofpartialderivativesandhowtoapplythemtosolveproblems.Somecommonstepstofollowinclude:</p><ul><li>Definethefunctionandthevariablesinvolved</li><li>Findthepartialderivativesofthefunctionwithrespecttoeachvariable</li><li>Usethepartialderivativestosolvetheproblem</li></ul><h3><strong>Q:Whataresomecommonmistakestoavoidwhenworkingwithpartialderivatives?</strong></h3><p>A:Somecommonmistakestoavoidwhenworkingwithpartialderivativesinclude:</p><ul><li>Failingtousethechainrule</li><li>Notsubstitutingthevalueofthevariableintotheequation</li><li>Notsimplifyingtheequation</li><li>Notcheckingtheunitsofthevariables</li></ul><p>Byfollowingthesestepsandavoidingcommonmistakes,youcanusepartialderivativestosolveawiderangeofproblemsinvariousfields.</p>\frac{\partial z}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x} </span></p> <h3><strong>Q: How do I calculate the partial derivative using the chain rule?</strong></h3> <p>A: To calculate the partial derivative using the chain rule, you need to find the derivatives of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(u)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>. We have:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac><mo>=</mo><mn>8</mn><msup><mi>u</mi><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial u} = 8u^7 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord">8</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span></p> <p>and</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">\frac{\partial u}{\partial x} = 6 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span></span></span></span></span></p> <p>Substituting these values into the chain rule formula, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>8</mn><msup><mi>u</mi><mn>7</mn></msup><mo>⋅</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">\frac{\partial z}{\partial x} = 8u^7 \cdot 6 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord">8</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span></span></span></span></span></p> <p>Now, we need to substitute the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> into the equation. We have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">u = 6x + 5y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>, so we can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>8</mn><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mo stretchy="false">)</mo><mn>7</mn></msup><mo>⋅</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">\frac{\partial z}{\partial x} = 8(6x + 5y)^7 \cdot 6 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">8</span><span class="mopen">(</span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span></span></span></span></span></p> <p>Simplifying the equation, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>48</mn><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mo stretchy="false">)</mo><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\frac{\partial z}{\partial x} = 48(6x + 5y)^7 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">48</span><span class="mopen">(</span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span></p> <h3><strong>Q: What is the partial derivative of the function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mo stretchy="false">)</mo><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">z = (6x + 5y)^8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span> with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>?</strong></h3> <p>A: To find the partial derivative of the function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mo stretchy="false">)</mo><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">z = (6x + 5y)^8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span> with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>, we will use the chain rule. The chain rule states that if we have a composite function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z = f(u)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u = g(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>, then the partial derivative of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span> with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> is given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial z}{\partial y} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p>In this case, we have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mo stretchy="false">)</mo><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">z = (6x + 5y)^8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span>, so we can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial z}{\partial y} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>u</mi><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">f(u) = u^8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">u = 6x + 5y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>.</p> <h3><strong>Q: How do I calculate the partial derivative with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>?</strong></h3> <p>A: To calculate the partial derivative with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>, we need to find the derivatives of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(u)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>. We have:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac><mo>=</mo><mn>8</mn><msup><mi>u</mi><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial u} = 8u^7 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord">8</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span></p> <p>and</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">\frac{\partial u}{\partial y} = 5 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span></span></p> <p>Substituting these values into the chain rule formula, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>8</mn><msup><mi>u</mi><mn>7</mn></msup><mo>⋅</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">\frac{\partial z}{\partial y} = 8u^7 \cdot 5 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord">8</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span></span></p> <p>Now, we need to substitute the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> into the equation. We have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">u = 6x + 5y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>, so we can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>8</mn><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mo stretchy="false">)</mo><mn>7</mn></msup><mo>⋅</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">\frac{\partial z}{\partial y} = 8(6x + 5y)^7 \cdot 5 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">8</span><span class="mopen">(</span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span></span></p> <p>Simplifying the equation, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>40</mn><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><msup><mo stretchy="false">)</mo><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\frac{\partial z}{\partial y} = 40(6x + 5y)^7 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">40</span><span class="mopen">(</span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span></p> <h3><strong>Q: What are some common applications of partial derivatives?</strong></h3> <p>A: Partial derivatives have many applications in various fields, including physics, engineering, economics, and computer science. Some common applications of partial derivatives include:</p> <ul> <li>Finding the maximum or minimum of a function</li> <li>Determining the rate of change of a function</li> <li>Solving optimization problems</li> <li>Modeling real-world phenomena</li> </ul> <h3><strong>Q: How do I use partial derivatives in real-world applications?</strong></h3> <p>A: To use partial derivatives in real-world applications, you need to understand the concept of partial derivatives and how to apply them to solve problems. Some common steps to follow include:</p> <ul> <li>Define the function and the variables involved</li> <li>Find the partial derivatives of the function with respect to each variable</li> <li>Use the partial derivatives to solve the problem</li> </ul> <h3><strong>Q: What are some common mistakes to avoid when working with partial derivatives?</strong></h3> <p>A: Some common mistakes to avoid when working with partial derivatives include:</p> <ul> <li>Failing to use the chain rule</li> <li>Not substituting the value of the variable into the equation</li> <li>Not simplifying the equation</li> <li>Not checking the units of the variables</li> </ul> <p>By following these steps and avoiding common mistakes, you can use partial derivatives to solve a wide range of problems in various fields.</p>