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:**
Suppose that
\[ f(x) = (4x - 4)^2 (3x^2 + 1)^3. \]
**Tasks:**
1. Find \( f'(x) \).
2. Evaluate \( f'(x) \) at \( x = 1 \) and \( x = -1 \).
---
**Solution:**
To solve this problem, you will need to use the product rule and chain rule for differentiation.
First, express \( f(x) \) as a product of two functions, \( u(x) = (4x - 4)^2 \) and \( v(x) = (3x^2 + 1)^3 \).
Differentiate each function:
- Find \( u'(x) \) using the chain rule.
- Find \( v'(x) \) also using the chain rule.
Apply the product rule:
\[ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x). \]
Next, evaluate \( f'(x) \) at the given points \( x = 1 \) and \( x = -1 \).
**Evaluation:**
- Find \( f'(1) \).
- Find \( f'(-1) \).
Once you compute the derivatives and substitute into the expressions, record the values:
\[ f'(1) = \text{(calculated value)} \]
\[ f'(-1) = \text{(calculated value)} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6ec190f-53e9-4e94-9a25-0c7f4fe861d8%2F9134db6d-5831-4311-8eb1-943d0567515b%2F21iy6md_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Suppose that
\[ f(x) = (4x - 4)^2 (3x^2 + 1)^3. \]
**Tasks:**
1. Find \( f'(x) \).
2. Evaluate \( f'(x) \) at \( x = 1 \) and \( x = -1 \).
---
**Solution:**
To solve this problem, you will need to use the product rule and chain rule for differentiation.
First, express \( f(x) \) as a product of two functions, \( u(x) = (4x - 4)^2 \) and \( v(x) = (3x^2 + 1)^3 \).
Differentiate each function:
- Find \( u'(x) \) using the chain rule.
- Find \( v'(x) \) also using the chain rule.
Apply the product rule:
\[ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x). \]
Next, evaluate \( f'(x) \) at the given points \( x = 1 \) and \( x = -1 \).
**Evaluation:**
- Find \( f'(1) \).
- Find \( f'(-1) \).
Once you compute the derivatives and substitute into the expressions, record the values:
\[ f'(1) = \text{(calculated value)} \]
\[ f'(-1) = \text{(calculated value)} \]
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