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:**
If \( f(x) = \left( x^2 + 5x + 3 \right)^4 \), then find the derivative \( f'(x) \).
**Solution:**
To find the derivative \( f'(x) \), use the chain rule.
Let \( u(x) = x^2 + 5x + 3 \).
Then, \( f(x) = u^4 \).
First, find the derivative of \( u(x) \):
\[
u'(x) = \frac{d}{dx}(x^2 + 5x + 3) = 2x + 5
\]
Now apply the chain rule for \( f(x) = u^4 \):
\[
f'(x) = \frac{d}{du}(u^4) \cdot u'(x) = 4u^3 \cdot (2x + 5)
\]
Substitute back \( u = x^2 + 5x + 3 \):
\[
f'(x) = 4(x^2 + 5x + 3)^3 \cdot (2x + 5)
\]
Thus, the derivative is:
\[
f'(x) = 4(x^2 + 5x + 3)^3 (2x + 5)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F437d091c-bafd-402f-b1ab-0058f5d0273f%2Fc1cc77a0-d338-49f1-ae3b-24f823565e29%2F8oib9js_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
If \( f(x) = \left( x^2 + 5x + 3 \right)^4 \), then find the derivative \( f'(x) \).
**Solution:**
To find the derivative \( f'(x) \), use the chain rule.
Let \( u(x) = x^2 + 5x + 3 \).
Then, \( f(x) = u^4 \).
First, find the derivative of \( u(x) \):
\[
u'(x) = \frac{d}{dx}(x^2 + 5x + 3) = 2x + 5
\]
Now apply the chain rule for \( f(x) = u^4 \):
\[
f'(x) = \frac{d}{du}(u^4) \cdot u'(x) = 4u^3 \cdot (2x + 5)
\]
Substitute back \( u = x^2 + 5x + 3 \):
\[
f'(x) = 4(x^2 + 5x + 3)^3 \cdot (2x + 5)
\]
Thus, the derivative is:
\[
f'(x) = 4(x^2 + 5x + 3)^3 (2x + 5)
\]
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