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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![**Problem Statement:**
Let \( f(x) = (2x^2 + 8x - 1)^4 \). Find \( f'(x) \).
**Solution:**
To find the derivative of \( f(x) \), we must use the chain rule. The chain rule states that if we have a composite function \( g(h(x)) \), then the derivative is \( g'(h(x)) \cdot h'(x) \).
In this case, let \( u = 2x^2 + 8x - 1 \). Thus, \( f(x) = u^4 \).
First, we find the derivative of \( u^4 \) with respect to \( u \):
\[ \frac{d}{du} (u^4) = 4u^3. \]
Next, we find the derivative of \( u \) with respect to \( x \):
\[ \frac{d}{dx} (2x^2 + 8x - 1) = 4x + 8. \]
Now using the chain rule, we combine these results:
\[ f'(x) = 4u^3 \cdot \frac{d}{dx}(2x^2 + 8x - 1). \]
Substitute back \( u = 2x^2 + 8x - 1 \):
\[ f'(x) = 4(2x^2 + 8x - 1)^3 \cdot (4x + 8). \]
So the derivative is:
\[ f'(x) = 4(2x^2 + 8x - 1)^3 (4x + 8). \]
Therefore, the answer is:
\[ f'(x) = 4(2x^2 + 8x - 1)^3 (4x + 8). \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa043fe4-4451-42e8-82d1-21cc59f9ce63%2F548c6793-92bf-4588-8e94-428dac1f4fd8%2Fn55pjoe.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( f(x) = (2x^2 + 8x - 1)^4 \). Find \( f'(x) \).
**Solution:**
To find the derivative of \( f(x) \), we must use the chain rule. The chain rule states that if we have a composite function \( g(h(x)) \), then the derivative is \( g'(h(x)) \cdot h'(x) \).
In this case, let \( u = 2x^2 + 8x - 1 \). Thus, \( f(x) = u^4 \).
First, we find the derivative of \( u^4 \) with respect to \( u \):
\[ \frac{d}{du} (u^4) = 4u^3. \]
Next, we find the derivative of \( u \) with respect to \( x \):
\[ \frac{d}{dx} (2x^2 + 8x - 1) = 4x + 8. \]
Now using the chain rule, we combine these results:
\[ f'(x) = 4u^3 \cdot \frac{d}{dx}(2x^2 + 8x - 1). \]
Substitute back \( u = 2x^2 + 8x - 1 \):
\[ f'(x) = 4(2x^2 + 8x - 1)^3 \cdot (4x + 8). \]
So the derivative is:
\[ f'(x) = 4(2x^2 + 8x - 1)^3 (4x + 8). \]
Therefore, the answer is:
\[ f'(x) = 4(2x^2 + 8x - 1)^3 (4x + 8). \]
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