Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
10)
![Find \( f' \) in terms of \( g' \)
\( f(x) = [g(x)]^3 \)
---
**Explanation for Educational Website:**
The problem involves finding the derivative \( f' \) of a function \( f(x) \) which is defined in terms of another function \( g(x) \). In this case, \( f(x) = [g(x)]^3 \).
To find \( f' \), you will need to use the chain rule from calculus. The chain rule states that the derivative of a composite function can be found by differentiating the outer function and then multiplying it by the derivative of the inner function.
Therefore, if \( f(x) = [g(x)]^3 \), then:
\[
f'(x) = 3[g(x)]^2 \cdot g'(x)
\]
This shows how \( f' \) can be expressed in terms of \( g' \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc92c6411-e0c2-4822-8067-9688589da753%2F80978689-e124-4309-8041-df2660c2aa45%2F5o87uhp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Find \( f' \) in terms of \( g' \)
\( f(x) = [g(x)]^3 \)
---
**Explanation for Educational Website:**
The problem involves finding the derivative \( f' \) of a function \( f(x) \) which is defined in terms of another function \( g(x) \). In this case, \( f(x) = [g(x)]^3 \).
To find \( f' \), you will need to use the chain rule from calculus. The chain rule states that the derivative of a composite function can be found by differentiating the outer function and then multiplying it by the derivative of the inner function.
Therefore, if \( f(x) = [g(x)]^3 \), then:
\[
f'(x) = 3[g(x)]^2 \cdot g'(x)
\]
This shows how \( f' \) can be expressed in terms of \( g' \).
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