Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
Q13
**Table of Values for \( f, g, f', \) and \( g' \)** The following table provides values for functions \( f(x) \), \( g(x) \), and their derivatives \( f'(x) \), \( g'(x) \) at specific points \( x \). | \( x \) | \( f(x) \) | \( g(x) \) | \( f'(x) \) | \( g'(x) \) | |--------|--------|--------|---------|---------| | -8 | -6 | -7 | -3 | -9 | | -7 | -8 | -1 | -4 | -2 | | -6 | -2 | -7 | -2 | 0 | **Problems** (a) If \( h(x) = f(g(x)) \), find \( h'(-8) \). \( h'(-8) = \_\_\_\_ \) (b) If \( H(x) = g(f(x)) \), find \( H'(-8) \). \( H'(-8) = \_\_\_\_ \) **Explanation of Problems** - Use the chain rule for differentiation to find the derivatives of the compositions. - For (a), calculate \( h'(-8) \) using \( h(x) = f(g(x)) \). - For (b), calculate \( H'(-8) \) using \( H(x) = g(f(x)) \).
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Transcribed Image Text:**Table of Values for \( f, g, f', \) and \( g' \)** The following table provides values for functions \( f(x) \), \( g(x) \), and their derivatives \( f'(x) \), \( g'(x) \) at specific points \( x \). | \( x \) | \( f(x) \) | \( g(x) \) | \( f'(x) \) | \( g'(x) \) | |--------|--------|--------|---------|---------| | -8 | -6 | -7 | -3 | -9 | | -7 | -8 | -1 | -4 | -2 | | -6 | -2 | -7 | -2 | 0 | **Problems** (a) If \( h(x) = f(g(x)) \), find \( h'(-8) \). \( h'(-8) = \_\_\_\_ \) (b) If \( H(x) = g(f(x)) \), find \( H'(-8) \). \( H'(-8) = \_\_\_\_ \) **Explanation of Problems** - Use the chain rule for differentiation to find the derivatives of the compositions. - For (a), calculate \( h'(-8) \) using \( h(x) = f(g(x)) \). - For (b), calculate \( H'(-8) \) using \( H(x) = g(f(x)) \).
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