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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2.4 #4
![**Question:**
If \( f(x) = \sin(x^3) \), find \( f'(x) \).
---
Find \( f'(1) \).
---
**Explanation:**
To solve the problem, we'll use the chain rule for differentiation. The function \( f(x) = \sin(x^3) \) is a composition of functions, where the outer function is \( \sin(u) \) and the inner function is \( u = x^3 \). The derivative \( f'(x) \) can be found by differentiating the outer function with respect to the inner function and then multiplying it by the derivative of the inner function.
**Steps:**
1. Differentiate the outer function \( \sin(u) \) with respect to \( u \) to get \( \cos(u) \).
2. Differentiate the inner function \( u = x^3 \) with respect to \( x \) to get \( 3x^2 \).
3. Apply the chain rule: \( f'(x) = \cos(x^3) \cdot 3x^2 \).
Thus, the derivative of \( f(x) \) is \( f'(x) = 3x^2 \cos(x^3) \).
To find \( f'(1) \):
Substitute \( x = 1 \) into \( f'(x) \):
\[ f'(1) = 3(1)^2 \cos(1^3) = 3 \cos(1) \].
So, \( f'(1) = 3 \cos(1) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e645a21-62be-4342-b7b9-aff3b926ed33%2F4d07090b-e927-4968-8e71-8a30ee851cbd%2F7dsj35f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
If \( f(x) = \sin(x^3) \), find \( f'(x) \).
---
Find \( f'(1) \).
---
**Explanation:**
To solve the problem, we'll use the chain rule for differentiation. The function \( f(x) = \sin(x^3) \) is a composition of functions, where the outer function is \( \sin(u) \) and the inner function is \( u = x^3 \). The derivative \( f'(x) \) can be found by differentiating the outer function with respect to the inner function and then multiplying it by the derivative of the inner function.
**Steps:**
1. Differentiate the outer function \( \sin(u) \) with respect to \( u \) to get \( \cos(u) \).
2. Differentiate the inner function \( u = x^3 \) with respect to \( x \) to get \( 3x^2 \).
3. Apply the chain rule: \( f'(x) = \cos(x^3) \cdot 3x^2 \).
Thus, the derivative of \( f(x) \) is \( f'(x) = 3x^2 \cos(x^3) \).
To find \( f'(1) \):
Substitute \( x = 1 \) into \( f'(x) \):
\[ f'(1) = 3(1)^2 \cos(1^3) = 3 \cos(1) \].
So, \( f'(1) = 3 \cos(1) \).
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