Differentiate The Given Function.${ Y = X(x^3 + 6)^4 }$

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Introduction

In calculus, differentiation is a fundamental concept that deals with the study of rates of change and slopes of curves. It is a crucial tool in various fields, including physics, engineering, and economics. In this article, we will focus on differentiating a given function, which is a polynomial function with a power of 4. We will use various techniques, including the chain rule and the product rule, to find the derivative of the function.

The Given Function

The given function is:

y=x(x3+6)4{ y = x(x^3 + 6)^4 }

This function is a polynomial function with a power of 4. It is a composite function, meaning that it is a combination of two or more functions. In this case, the function is a product of two functions: x{ x } and (x3+6)4{ (x^3 + 6)^4 }.

Differentiating the Function

To differentiate the function, we will use the product rule, which states that if we have a function of the form f(x)=u(x)v(x){ f(x) = u(x)v(x) }, then the derivative of the function is given by:

f′(x)=u′(x)v(x)+u(x)v′(x){ f'(x) = u'(x)v(x) + u(x)v'(x) }

In this case, we have:

u(x)=x{ u(x) = x } v(x)=(x3+6)4{ v(x) = (x^3 + 6)^4 }

Using the product rule, we can find the derivative of the function as follows:

{ y' = (x)'(x^3 + 6)^4 + x(x^3 + 6)^4' }

Now, we need to find the derivatives of u(x){ u(x) } and v(x){ v(x) }. The derivative of u(x){ u(x) } is simply 1, since the derivative of x{ x } is 1. The derivative of v(x){ v(x) } is:

v′(x)=4(x3+6)3(3x2){ v'(x) = 4(x^3 + 6)^3(3x^2) }

Now, we can substitute the derivatives of u(x){ u(x) } and v(x){ v(x) } into the product rule formula:

y′=1(x3+6)4+x(4(x3+6)3(3x2)){ y' = 1(x^3 + 6)^4 + x(4(x^3 + 6)^3(3x^2)) }

Simplifying the expression, we get:

y′=(x3+6)4+12x3(x3+6)3{ y' = (x^3 + 6)^4 + 12x^3(x^3 + 6)^3 }

Simplifying the Derivative

We can simplify the derivative further by combining like terms. We can factor out (x3+6)3{ (x^3 + 6)^3 } from both terms:

y′=(x3+6)3((x3+6)+12x3){ y' = (x^3 + 6)^3((x^3 + 6) + 12x^3) }

Simplifying the expression inside the parentheses, we get:

y′=(x3+6)3(13x3+6){ y' = (x^3 + 6)^3(13x^3 + 6) }

Conclusion

In this article, we differentiated a given function using the product rule. We found the derivative of the function by applying the product rule and simplifying the expression. The derivative of the function is:

y′=(x3+6)3(13x3+6){ y' = (x^3 + 6)^3(13x^3 + 6) }

This derivative can be used to find the rate of change of the function at any point. It can also be used to find the slope of the tangent line to the curve at any point.

Applications of Differentiation

Differentiation has many applications in various fields, including physics, engineering, and economics. Some of the applications of differentiation include:

  • Physics: Differentiation is used to describe the motion of objects. It is used to find the velocity and acceleration of an object at any point in time.
  • Engineering: Differentiation is used to design and optimize systems. It is used to find the maximum and minimum values of a function, which is essential in designing systems that meet certain criteria.
  • Economics: Differentiation is used to model economic systems. It is used to find the rate of change of economic variables, such as the rate of inflation or the rate of unemployment.

Final Thoughts

In conclusion, differentiation is a powerful tool that has many applications in various fields. It is used to find the rate of change of a function at any point, which is essential in understanding the behavior of a function. In this article, we differentiated a given function using the product rule and simplified the expression. The derivative of the function is:

y′=(x3+6)3(13x3+6){ y' = (x^3 + 6)^3(13x^3 + 6) }

This derivative can be used to find the rate of change of the function at any point. It can also be used to find the slope of the tangent line to the curve at any point.

References

  • Calculus: A First Course by Michael Sullivan
  • Calculus: Early Transcendentals by James Stewart
  • Calculus: Single Variable by Michael Spivak

Glossary

  • Derivative: A measure of the rate of change of a function at a point.
  • Product Rule: A rule for differentiating a product of two functions.
  • Chain Rule: A rule for differentiating a composite function.
  • Composite Function: A function that is a combination of two or more functions.
    Differentiating the Given Function: A Comprehensive Approach ===========================================================

Q&A: Differentiating the Given Function

Q: What is the given function?

A: The given function is:

y=x(x3+6)4{ y = x(x^3 + 6)^4 }

Q: What is the derivative of the given function?

A: The derivative of the given function is:

y′=(x3+6)3(13x3+6){ y' = (x^3 + 6)^3(13x^3 + 6) }

Q: How did you find the derivative of the given function?

A: We used the product rule to find the derivative of the given function. The product rule states that if we have a function of the form f(x)=u(x)v(x){ f(x) = u(x)v(x) }, then the derivative of the function is given by:

f′(x)=u′(x)v(x)+u(x)v′(x){ f'(x) = u'(x)v(x) + u(x)v'(x) }

In this case, we have:

u(x)=x{ u(x) = x } v(x)=(x3+6)4{ v(x) = (x^3 + 6)^4 }

Using the product rule, we can find the derivative of the function as follows:

{ y' = (x)'(x^3 + 6)^4 + x(x^3 + 6)^4' }

Q: What is the product rule?

A: The product rule is a rule for differentiating a product of two functions. It states that if we have a function of the form f(x)=u(x)v(x){ f(x) = u(x)v(x) }, then the derivative of the function is given by:

f′(x)=u′(x)v(x)+u(x)v′(x){ f'(x) = u'(x)v(x) + u(x)v'(x) }

Q: What is the chain rule?

A: The chain rule is a rule for differentiating a composite function. It states that if we have a function of the form f(x)=g(h(x)){ f(x) = g(h(x)) }, then the derivative of the function is given by:

f′(x)=g′(h(x))⋅h′(x){ f'(x) = g'(h(x)) \cdot h'(x) }

Q: How is the chain rule related to the product rule?

A: The chain rule is related to the product rule in that it can be used to differentiate composite functions. The product rule can be used to differentiate products of two functions, while the chain rule can be used to differentiate composite functions.

Q: What are some applications of differentiation?

A: Differentiation has many applications in various fields, including physics, engineering, and economics. Some of the applications of differentiation include:

  • Physics: Differentiation is used to describe the motion of objects. It is used to find the velocity and acceleration of an object at any point in time.
  • Engineering: Differentiation is used to design and optimize systems. It is used to find the maximum and minimum values of a function, which is essential in designing systems that meet certain criteria.
  • Economics: Differentiation is used to model economic systems. It is used to find the rate of change of economic variables, such as the rate of inflation or the rate of unemployment.

Q: What are some common mistakes to avoid when differentiating functions?

A: Some common mistakes to avoid when differentiating functions include:

  • Forgetting to apply the product rule: When differentiating a product of two functions, it is essential to apply the product rule.
  • Forgetting to apply the chain rule: When differentiating a composite function, it is essential to apply the chain rule.
  • Not simplifying the derivative: After finding the derivative of a function, it is essential to simplify the expression to make it easier to work with.

Q: How can I practice differentiating functions?

A: There are many ways to practice differentiating functions, including:

  • Solving problems: Practice solving problems that involve differentiating functions.
  • Using online resources: There are many online resources available that can help you practice differentiating functions, including video tutorials and practice problems.
  • Working with a tutor: Working with a tutor can be a great way to practice differentiating functions and get feedback on your work.

Q: What are some tips for differentiating functions?

A: Some tips for differentiating functions include:

  • Read the problem carefully: Before starting to differentiate a function, read the problem carefully to make sure you understand what is being asked.
  • Apply the product rule and chain rule: When differentiating a product of two functions or a composite function, apply the product rule and chain rule, respectively.
  • Simplify the derivative: After finding the derivative of a function, simplify the expression to make it easier to work with.