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Suppose that S(m)3(4#-4)*(32+ 1). Find f'(#), and then evaluate f' at # = 1 and # = -1 (1)= 1(-1)=

Suppose that S(m)3(4#-4)*(32+ 1). Find f'(#), and then evaluate f' at # = 1 and # = -1 (1)= 1(-1)=

Calculus: Early Transcendentals
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:** 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: \[ f'(1) = \text{(calculated value)} \] \[ f'(-1) = \text{(calculated value)} \]
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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