Use The Chain Rule To Find The Derivative Of ${ F(x) = 2\left(-2 X^6 + 5 X 8\right) {15} } Y O U D O N O T N E E D T O E X P A N D O U T Y O U R A N S W E R . You Do Not Need To Expand Out Your Answer. Y O U D O N O T N Ee D T Oe X P An D O U T Yo U R An S W Er . { F^{\prime}(x) = \}

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Introduction

In calculus, the chain rule is a fundamental concept that helps us find the derivative of composite functions. It's a powerful tool that allows us to differentiate functions that are composed of other functions. In this article, we'll explore how to use the chain rule to find the derivative of a given function, specifically the function f(x)=2(βˆ’2x6+5x8)15f(x) = 2\left(-2 x^6 + 5 x^8\right)^{15}.

What is the Chain Rule?

The chain rule is a formula that helps us find the derivative of a composite function. It states that if we have a function of the form f(x)=g(h(x))f(x) = g(h(x)), where gg and hh are both functions of xx, then the derivative of ff with respect to xx is given by:

fβ€²(x)=gβ€²(h(x))β‹…hβ€²(x)f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)

Applying the Chain Rule to the Given Function

Now that we've introduced the chain rule, let's apply it to the given function f(x)=2(βˆ’2x6+5x8)15f(x) = 2\left(-2 x^6 + 5 x^8\right)^{15}. To do this, we need to identify the outer function gg and the inner function hh.

In this case, the outer function gg is g(u)=2u15g(u) = 2u^{15}, where u=βˆ’2x6+5x8u = -2 x^6 + 5 x^8. The inner function hh is h(x)=βˆ’2x6+5x8h(x) = -2 x^6 + 5 x^8.

Finding the Derivative of the Outer Function

To find the derivative of the outer function gg, we'll use the power rule of differentiation. The power rule states that if we have a function of the form f(x)=xnf(x) = x^n, then the derivative of ff with respect to xx is given by:

fβ€²(x)=nxnβˆ’1f^{\prime}(x) = n x^{n-1}

In this case, we have g(u)=2u15g(u) = 2u^{15}, so the derivative of gg with respect to uu is:

gβ€²(u)=30u14g^{\prime}(u) = 30 u^{14}

Finding the Derivative of the Inner Function

To find the derivative of the inner function hh, we'll use the power rule of differentiation again. We have h(x)=βˆ’2x6+5x8h(x) = -2 x^6 + 5 x^8, so the derivative of hh with respect to xx is:

hβ€²(x)=βˆ’12x5+40x7h^{\prime}(x) = -12 x^5 + 40 x^7

Applying the Chain Rule

Now that we've found the derivatives of the outer and inner functions, we can apply the chain rule to find the derivative of the given function ff. We have:

fβ€²(x)=gβ€²(h(x))β‹…hβ€²(x)f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)

Substituting the expressions we found earlier, we get:

fβ€²(x)=30(βˆ’2x6+5x8)14β‹…(βˆ’12x5+40x7)f^{\prime}(x) = 30 \left(-2 x^6 + 5 x^8\right)^{14} \cdot \left(-12 x^5 + 40 x^7\right)

Simplifying the Expression

We can simplify the expression by combining like terms. We have:

fβ€²(x)=30(βˆ’2x6+5x8)14β‹…(βˆ’12x5+40x7)f^{\prime}(x) = 30 \left(-2 x^6 + 5 x^8\right)^{14} \cdot \left(-12 x^5 + 40 x^7\right)

fβ€²(x)=30(βˆ’2x6+5x8)14β‹…(βˆ’12x5+40x7)f^{\prime}(x) = 30 \left(-2 x^6 + 5 x^8\right)^{14} \cdot \left(-12 x^5 + 40 x^7\right)

fβ€²(x)=30(βˆ’2x6+5x8)14β‹…(βˆ’12x5+40x7)f^{\prime}(x) = 30 \left(-2 x^6 + 5 x^8\right)^{14} \cdot \left(-12 x^5 + 40 x^7\right)

Conclusion

In this article, we used the chain rule to find the derivative of the given function f(x)=2(βˆ’2x6+5x8)15f(x) = 2\left(-2 x^6 + 5 x^8\right)^{15}. We identified the outer function gg and the inner function hh, found the derivatives of both functions, and applied the chain rule to find the derivative of ff. The final expression for the derivative of ff is:

fβ€²(x)=30(βˆ’2x6+5x8)14β‹…(βˆ’12x5+40x7)f^{\prime}(x) = 30 \left(-2 x^6 + 5 x^8\right)^{14} \cdot \left(-12 x^5 + 40 x^7\right)

Final Answer

f^{\prime}(x) = 30 \left(-2 x^6 + 5 x^8\right)^{14} \cdot \left(-12 x^5 + 40 x^7\right)$<br/> **The Chain Rule in Calculus: A Q&A Guide** =====================================================

Introduction

In our previous article, we explored how to use the chain rule to find the derivative of a composite function. The chain rule is a powerful tool that allows us to differentiate functions that are composed of other functions. In this article, we'll answer some common questions about the chain rule and provide additional examples to help you understand this concept better.

Q: What is the chain rule?

A: The chain rule is a formula that helps us find the derivative of a composite function. It states that if we have a function of the form f(x)=g(h(x))f(x) = g(h(x)), where gg and hh are both functions of xx, then the derivative of ff with respect to xx is given by:

fβ€²(x)=gβ€²(h(x))β‹…hβ€²(x)</span></p><h2><strong>Q:WhendoIusethechainrule?</strong></h2><p>A:Youusethechainrulewhenyouhaveacompositefunction,meaningafunctionthatiscomposedofotherfunctions.Forexample,ifyouhaveafunctionlike<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁑</mo><mostretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mostretchy="false">)</mo></mrow><annotationencoding="application/xβˆ’tex">f(x)=sin⁑(x2)</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0641em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mop">sin</span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:βˆ’3.063em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mclose">)</span></span></span></span>,youwouldusethechainruletofinditsderivative.</p><h2><strong>Q:HowdoIapplythechainrule?</strong></h2><p>A:Toapplythechainrule,youneedtoidentifytheouterfunction<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotationencoding="application/xβˆ’tex">g</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;verticalβˆ’align:βˆ’0.1944em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.03588em;">g</span></span></span></span>andtheinnerfunction<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotationencoding="application/xβˆ’tex">h</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">h</span></span></span></span>.Then,youfindthederivativesofbothfunctionsandmultiplythemtogether.Thefinalexpressionforthederivativeofthecompositefunctionis:</p><pclass=β€²katexβˆ’blockβ€²><spanclass="katexβˆ’display"><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msup><mi>f</mi><momathvariant="normal"lspace="0em"rspace="0em">β€²</mo></msup><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><msup><mi>g</mi><momathvariant="normal"lspace="0em"rspace="0em">β€²</mo></msup><mostretchy="false">(</mo><mi>h</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mostretchy="false">)</mo><mo>β‹…</mo><msup><mi>h</mi><momathvariant="normal"lspace="0em"rspace="0em">β€²</mo></msup><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/xβˆ’tex">fβ€²(x)=gβ€²(h(x))β‹…hβ€²(x)</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.0519em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="marginβˆ’right:0.10764em;">f</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8019em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">β€²</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0519em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="marginβˆ’right:0.03588em;">g</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8019em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">β€²</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">h</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">))</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">β‹…</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0519em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord"><spanclass="mordmathnormal">h</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8019em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">β€²</span></span></span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span></span></p><h2><strong>Q:WhatifIhaveafunctionwithmultiplelayersofcomposition?</strong></h2><p>A:Ifyouhaveafunctionwithmultiplelayersofcomposition,youcanstillusethechainrule.Forexample,ifyouhaveafunctionlike<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁑</mo><mostretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/xβˆ’tex">f(x)=sin⁑(x2+3x)</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0641em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mop">sin</span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:βˆ’3.063em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mord">3</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span>,youwouldfirstfindthederivativeoftheinnerfunction<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi></mrow><annotationencoding="application/xβˆ’tex">x2+3x</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8974em;verticalβˆ’align:βˆ’0.0833em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:βˆ’3.063em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span><spanclass="mordmathnormal">x</span></span></span></span>,andthenusethechainruletofindthederivativeoftheouterfunction<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁑</mo><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/xβˆ’tex">sin⁑(x)</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mop">sin</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span>.</p><h2><strong>Q:CanIusethechainrulewithothertypesoffunctions?</strong></h2><p>A:Yes,youcanusethechainrulewithothertypesoffunctions,suchasexponentialfunctions,logarithmicfunctions,andtrigonometricfunctions.Thechainruleisageneralrulethatappliestoanycompositefunction.</p><h2><strong>Q:HowdoIknowwhentousethechainruleandwhentousetheproductrule?</strong></h2><p>A:Thechainruleandtheproductrulearebothusedtofindthederivativeofafunction,buttheyareusedindifferentsituations.Thechainruleisusedwhenyouhaveacompositefunction,whiletheproductruleisusedwhenyouhaveaproductoftwofunctions.Ifyouβ€²renotsurewhichruletouse,trytoidentifythetypeoffunctionyouβ€²reworkingwithandusethecorrespondingrule.</p><h2><strong>Q:CanIusethechainrulewithimplicitdifferentiation?</strong></h2><p>A:Yes,youcanusethechainrulewithimplicitdifferentiation.Implicitdifferentiationisatechniqueusedtofindthederivativeofafunctionthatisdefinedimplicitly,meaningthatitisdefinedintermsofanotherfunction.Thechainruleisusedtofindthederivativeoftheimplicitfunction.</p><h2><strong>Conclusion</strong></h2><p>Inthisarticle,weansweredsomecommonquestionsaboutthechainruleandprovidedadditionalexamplestohelpyouunderstandthisconceptbetter.Thechainruleisapowerfultoolthatallowsustodifferentiatefunctionsthatarecomposedofotherfunctions.Byunderstandinghowtousethechainrule,youβ€²llbeabletofindthederivativeofawiderangeoffunctionsandsolveavarietyofproblemsincalculus.</p><h2><strong>FinalTips</strong></h2><ul><li>Makesuretoidentifytheouterfunction<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotationencoding="application/xβˆ’tex">g</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;verticalβˆ’align:βˆ’0.1944em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.03588em;">g</span></span></span></span>andtheinnerfunction<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotationencoding="application/xβˆ’tex">h</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">h</span></span></span></span>whenapplyingthechainrule.</li><li>Findthederivativesofbothfunctionsandmultiplythemtogether.</li><li>Usethechainrulewithothertypesoffunctions,suchasexponentialfunctions,logarithmicfunctions,andtrigonometricfunctions.</li><li>Becarefulwhenusingthechainrulewithimplicitdifferentiation.</li></ul><h2><strong>CommonMistakestoAvoid</strong></h2><ul><li>Failingtoidentifytheouterfunction<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotationencoding="application/xβˆ’tex">g</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;verticalβˆ’align:βˆ’0.1944em;"></span><spanclass="mordmathnormal"style="marginβˆ’right:0.03588em;">g</span></span></span></span>andtheinnerfunction<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotationencoding="application/xβˆ’tex">h</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">h</span></span></span></span>.</li><li>Notfindingthederivativesofbothfunctions.</li><li>Notmultiplyingthederivativestogether.</li><li>Usingthechainrulewiththewrongtypeoffunction.</li></ul><h2><strong>AdditionalResources</strong></h2><ul><li>Formoreinformationonthechainrule,checkoutourpreviousarticleonthetopic.</li><li>Forpracticeproblemsandexamples,tryusingonlineresourcessuchasKhanAcademyorMITOpenCourseWare.</li><li>Foramoreinβˆ’depthunderstandingofthechainrule,tryreadingacalculustextbookortakingacalculuscourse.</li></ul>f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x) </span></p> <h2><strong>Q: When do I use the chain rule?</strong></h2> <p>A: You use the chain rule when you have a composite function, meaning a function that is composed of other functions. For example, if you have a function like <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>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁑</mo><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = \sin(x^2)</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">x</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:1.0641em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</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">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, you would use the chain rule to find its derivative.</p> <h2><strong>Q: How do I apply the chain rule?</strong></h2> <p>A: To apply the chain rule, you need to identify the outer function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</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;">g</span></span></span></span> and the inner function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>. Then, you find the derivatives of both functions and multiply them together. The final expression for the derivative of the composite function is:</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><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">β€²</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">β€²</mo></msup><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>β‹…</mo><msup><mi>h</mi><mo mathvariant="normal" lspace="0em" rspace="0em">β€²</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><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"><span class="mord mtight">β€²</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</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:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><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"><span class="mord mtight">β€²</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β‹…</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><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"><span class="mord mtight">β€²</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <h2><strong>Q: What if I have a function with multiple layers of composition?</strong></h2> <p>A: If you have a function with multiple layers of composition, you can still use the chain rule. For example, if you have a function like <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>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁑</mo><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = \sin(x^2 + 3x)</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">x</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:1.0641em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</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">2</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:1em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>, you would first find the derivative of the inner function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">x^2 + 3x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</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">2</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">3</span><span class="mord mathnormal">x</span></span></span></span>, and then use the chain rule to find the derivative of the outer function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁑</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sin(x)</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="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>.</p> <h2><strong>Q: Can I use the chain rule with other types of functions?</strong></h2> <p>A: Yes, you can use the chain rule with other types of functions, such as exponential functions, logarithmic functions, and trigonometric functions. The chain rule is a general rule that applies to any composite function.</p> <h2><strong>Q: How do I know when to use the chain rule and when to use the product rule?</strong></h2> <p>A: The chain rule and the product rule are both used to find the derivative of a function, but they are used in different situations. The chain rule is used when you have a composite function, while the product rule is used when you have a product of two functions. If you're not sure which rule to use, try to identify the type of function you're working with and use the corresponding rule.</p> <h2><strong>Q: Can I use the chain rule with implicit differentiation?</strong></h2> <p>A: Yes, you can use the chain rule with implicit differentiation. Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly, meaning that it is defined in terms of another function. The chain rule is used to find the derivative of the implicit function.</p> <h2><strong>Conclusion</strong></h2> <p>In this article, we answered some common questions about the chain rule and provided additional examples to help you understand this concept better. The chain rule is a powerful tool that allows us to differentiate functions that are composed of other functions. By understanding how to use the chain rule, you'll be able to find the derivative of a wide range of functions and solve a variety of problems in calculus.</p> <h2><strong>Final Tips</strong></h2> <ul> <li>Make sure to identify the outer function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</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;">g</span></span></span></span> and the inner function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> when applying the chain rule.</li> <li>Find the derivatives of both functions and multiply them together.</li> <li>Use the chain rule with other types of functions, such as exponential functions, logarithmic functions, and trigonometric functions.</li> <li>Be careful when using the chain rule with implicit differentiation.</li> </ul> <h2><strong>Common Mistakes to Avoid</strong></h2> <ul> <li>Failing to identify the outer function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</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;">g</span></span></span></span> and the inner function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>.</li> <li>Not finding the derivatives of both functions.</li> <li>Not multiplying the derivatives together.</li> <li>Using the chain rule with the wrong type of function.</li> </ul> <h2><strong>Additional Resources</strong></h2> <ul> <li>For more information on the chain rule, check out our previous article on the topic.</li> <li>For practice problems and examples, try using online resources such as Khan Academy or MIT OpenCourseWare.</li> <li>For a more in-depth understanding of the chain rule, try reading a calculus textbook or taking a calculus course.</li> </ul>