Identify the outer and inner functions

Oct 13, 2025 by magicalceylontours.com 31 views

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. When dealing with composite functions, we need to apply the chain rule to find the derivative. In this article, we will explore how to find the derivative of a composite function using the chain rule.

The Chain Rule

The chain rule is a fundamental concept in calculus that allows us to find the derivative of a composite function. It states that if we have a composite function of the form:

${ y = f(g(x)) }$

then the derivative of y with respect to x is given by:

${ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) }$

where f'(g(x)) is the derivative of f with respect to its argument, and g'(x) is the derivative of g with respect to x.

Applying the Chain Rule to the Given Function

Now, let's apply the chain rule to the given function:

${ y = -4\left(6 x^2 + 7 x\right)^{\frac{5}{2}} }$

To find the derivative of y with respect to x, we need to identify the outer and inner functions. In this case, the outer function is:

${ f(u) = -4u^{\frac{5}{2}} }$

where u is the inner function:

${ u = 6 x^2 + 7 x }$

Finding the Derivative of the Outer Function

To find the derivative of the outer function f(u), we can use the power rule:

${ f'(u) = -4 \cdot \frac{5}{2} u^{\frac{3}{2}} }$

${ f'(u) = -10 u^{\frac{3}{2}} }$

Finding the Derivative of the Inner Function

To find the derivative of the inner function u, we can use the sum rule and the power rule:

${ u' = \frac{d}{dx} (6 x^2 + 7 x) }$

${ u' = 12 x + 7 }$

Applying the Chain Rule

Now that we have found the derivatives of the outer and inner functions, we can apply the chain rule:

${ \frac{dy}{dx} = f'(u) \cdot u' }$

${ \frac{dy}{dx} = -10 u^{\frac{3}{2}} \cdot (12 x + 7) }$

Simplifying the Derivative

To simplify the derivative, we can substitute the expression for u:

${ u = 6 x^2 + 7 x }$

${ u^{\frac{3}{2}} = (6 x^2 + 7 x)^{\frac{3}{2}} }$

${ \frac{dy}{dx} = -10 (6 x^2 + 7 x)^{\frac{3}{2}} \cdot (12 x + 7) }$

Final Answer

The derivative of the given function is:

${ \frac{dy}{dx} = -10 (6 x^2 + 7 x)^{\frac{3}{2}} \cdot (12 x + 7) }$

This is the final answer.

Conclusion

In this article, we have applied the chain rule to find the derivative of a composite function. We have identified the outer and inner functions, found their derivatives, and applied the chain rule to obtain the final derivative. This technique is essential in calculus and has numerous applications in physics, engineering, and other fields.

Example Use Cases

The chain rule has numerous applications in various fields, including:

Physics: To find the velocity and acceleration of an object, we need to find the derivative of its position function.
Engineering: To design and optimize systems, we need to find the derivative of their performance functions.
Economics: To analyze the behavior of economic systems, we need to find the derivative of their functions.

Tips and Tricks

When applying the chain rule, make sure to:

Identify the outer and inner functions: This is crucial in applying the chain rule correctly.
Find the derivatives of the outer and inner functions: Use the power rule, sum rule, and product rule to find the derivatives.
Apply the chain rule: Multiply the derivatives of the outer and inner functions to obtain the final derivative.

By following these tips and tricks, you can master the chain rule and apply it to a wide range of problems in calculus and beyond.

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Q: What is the chain rule in calculus?

A: The chain rule is a fundamental concept in calculus that allows us to find the derivative of a composite function. It states that if we have a composite function of the form:

${ y = f(g(x)) }$

then the derivative of y with respect to x is given by:

${ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) }$

Q: How do I identify the outer and inner functions in a composite function?

A: To identify the outer and inner functions, look for the function that is being composed. The outer function is the function that is being applied to the inner function. For example, in the function:

${ y = -4\left(6 x^2 + 7 x\right)^{\frac{5}{2}} }$

the outer function is:

${ f(u) = -4u^{\frac{5}{2}} }$

and the inner function is:

${ u = 6 x^2 + 7 x }$

Q: How do I find the derivative of the outer function?

A: To find the derivative of the outer function, use the power rule:

${ f'(u) = -4 \cdot \frac{5}{2} u^{\frac{3}{2}} }$

${ f'(u) = -10 u^{\frac{3}{2}} }$

Q: How do I find the derivative of the inner function?

A: To find the derivative of the inner function, use the sum rule and the power rule:

${ u' = \frac{d}{dx} (6 x^2 + 7 x) }$

${ u' = 12 x + 7 }$

Q: How do I apply the chain rule to find the derivative of a composite function?

A: To apply the chain rule, multiply the derivatives of the outer and inner functions:

${ \frac{dy}{dx} = f'(u) \cdot u' }$

${ \frac{dy}{dx} = -10 u^{\frac{3}{2}} \cdot (12 x + 7) }$

Q: What are some common mistakes to avoid when applying the chain rule?

A: Some common mistakes to avoid when applying the chain rule include:

Not identifying the outer and inner functions correctly: Make sure to identify the outer and inner functions correctly before applying the chain rule.
Not finding the derivatives of the outer and inner functions correctly: Use the power rule, sum rule, and product rule to find the derivatives of the outer and inner functions.
Not applying the chain rule correctly: Multiply the derivatives of the outer and inner functions to obtain the final derivative.

Q: What are some real-world applications of the chain rule?

A: The chain rule has numerous real-world applications, including:

Physics: To find the velocity and acceleration of an object, we need to find the derivative of its position function.
Engineering: To design and optimize systems, we need to find the derivative of their performance functions.
Economics: To analyze the behavior of economic systems, we need to find the derivative of their functions.

Q: How can I practice applying the chain rule?

A: To practice applying the chain rule, try the following:

Work through examples: Practice finding the derivatives of composite functions using the chain rule.
Use online resources: There are many online resources available that provide practice problems and examples of applying the chain rule.
Seek help from a tutor or teacher: If you are struggling to apply the chain rule, seek help from a tutor or teacher.

By following these tips and practicing applying the chain rule, you can master this fundamental concept in calculus and apply it to a wide range of problems in physics, engineering, economics, and beyond.