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
![**Problem Statement:**
Given the function \( f(x) = (4x + 1)^5 \), compute the second derivative \( f''(x) \) at \( x = 0 \).
**Solution Approach:**
1. **Function Definition:**
\[
f(x) = (4x + 1)^5
\]
2. **First Derivative (Using Chain Rule):**
Let \( u = 4x + 1 \), then \( f(x) = u^5 \).
\[
\frac{du}{dx} = 4
\]
\[
\frac{df}{du} = 5u^4
\]
Applying the chain rule:
\[
\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = 5(4x + 1)^4 \cdot 4 = 20(4x + 1)^4
\]
3. **Second Derivative:**
Differentiate \( f'(x) = 20(4x + 1)^4 \) again using the chain rule.
\[
\frac{d}{dx}[(4x + 1)^4] = 4 \cdot 4(4x + 1)^3 = 16(4x + 1)^3
\]
So,
\[
f''(x) = 20 \cdot 16(4x + 1)^3 = 320(4x + 1)^3
\]
4. **Evaluate at \( x = 0 \):**
\[
f''(0) = 320(4(0) + 1)^3 = 320 \cdot 1^3 = 320
\]
**Answer:**
\( f''(0) = 320 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a969483-3cd4-4c69-a19f-4246f91513cf%2Fff27afa7-b7ce-4880-9ed9-ddf8ea93a9c1%2Fduxs8z_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the function \( f(x) = (4x + 1)^5 \), compute the second derivative \( f''(x) \) at \( x = 0 \).
**Solution Approach:**
1. **Function Definition:**
\[
f(x) = (4x + 1)^5
\]
2. **First Derivative (Using Chain Rule):**
Let \( u = 4x + 1 \), then \( f(x) = u^5 \).
\[
\frac{du}{dx} = 4
\]
\[
\frac{df}{du} = 5u^4
\]
Applying the chain rule:
\[
\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = 5(4x + 1)^4 \cdot 4 = 20(4x + 1)^4
\]
3. **Second Derivative:**
Differentiate \( f'(x) = 20(4x + 1)^4 \) again using the chain rule.
\[
\frac{d}{dx}[(4x + 1)^4] = 4 \cdot 4(4x + 1)^3 = 16(4x + 1)^3
\]
So,
\[
f''(x) = 20 \cdot 16(4x + 1)^3 = 320(4x + 1)^3
\]
4. **Evaluate at \( x = 0 \):**
\[
f''(0) = 320(4(0) + 1)^3 = 320 \cdot 1^3 = 320
\]
**Answer:**
\( f''(0) = 320 \)
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