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2. Each of the following "derivatives" has been done incorrectly. In each case below, find a coun- terexample that shows why the reasoning is incorrect: Then correct the mistakes by finding the actual derivative of the left hand side. For example: The statement "f(2x) = f' (2x)" is false because if d d f(x) = x, then we have f(2x) = [4.x²] = 8x, dx dx but on the other hand, f'(x) = 2x, so f'(2x) = 4x. In general, the correct derivative would be d f(2x) = f'(2x) · 2" dx by the chain rule. f(x+ 5) – f'(x) (b) 9(t/3) – gʻ(t)/3

2. Each of the following "derivatives" has been done incorrectly. In each case below, find a coun- terexample that shows why the reasoning is incorrect: Then correct the mistakes by finding the actual derivative of the left hand side. For example: The statement "f(2x) = f' (2x)" is false because if d d f(x) = x, then we have f(2x) = [4.x²] = 8x, dx dx but on the other hand, f'(x) = 2x, so f'(2x) = 4x. In general, the correct derivative would be d f(2x) = f'(2x) · 2" dx by the chain rule. f(x+ 5) – f'(x) (b) 9(t/3) – gʻ(t)/3

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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2. Each of the following "derivatives" has been done incorrectly. In each case below, find a coun- terexample that shows why the reasoning is incorrect: Then correct the mistakes by finding the actual derivative of the left hand side. For example: The statement "f(2x) = f' (2x)" is false because if d d. f(x) = x, then we have f(2x) = [4.02] = 8x, dx dx but on the other hand, f'(x) = 2x, so f'(2x) = 4x. In general, the correct derivative would be d f(2x) = f'(2x) · 2" dx by the chain rule, (a) (z+ 5) - r'(=) (b) 79(t/3) g'(t)/3 (c) k(=) K (2) (d) (F(u) sin(2u)] 2 2F'(u) cos(2u)
Transcribed Image Text:2. Each of the following "derivatives" has been done incorrectly. In each case below, find a coun- terexample that shows why the reasoning is incorrect: Then correct the mistakes by finding the actual derivative of the left hand side. For example: The statement "f(2x) = f' (2x)" is false because if d d. f(x) = x, then we have f(2x) = [4.02] = 8x, dx dx but on the other hand, f'(x) = 2x, so f'(2x) = 4x. In general, the correct derivative would be d f(2x) = f'(2x) · 2" dx by the chain rule, (a) (z+ 5) - r'(=) (b) 79(t/3) g'(t)/3 (c) k(=) K (2) (d) (F(u) sin(2u)] 2 2F'(u) cos(2u)
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