Explanation Formula used: The Chain Rule: If g is differentiable at x and f is differentiable at g ( x ) , then the composite function y = f ∘ g defined by y ( x ) = f ( g ( x ) ) is differentiable at x and y ′ is given by the product, y ′ = f ′ ( g ( x ) ) ⋅ g ′ ( x ) . Calculation: From the above chain rule, it is observed that if y = f ( g ( x ) ) then y ′ = f ′ ( g ( x ) ) ⋅ g ′ ( x ) . Therefore, if y = f ( g ( x ) ) then y ′ ≠ f ′ ( g ′ ( x ) ) . Counter Example: Obtain the differentiation of y = ( x 2 + 3 ) 3 . Let the inner function be u = g ( x ) and the outer function be y = f ( u ) . Here, g ( x ) = x 2 + 3 and f ( u ) = x 3

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

8th Edition

ISBN: 9781305279148

Author: Stewart, James, St. Andre, Richard

Publisher: Cengage Learning

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Ch. 3.1 - Prob. 1PT Ch. 3.1 - Prob. 2PT Ch. 3.1 - Prob. 3PT Ch. 3.1 - Prob. 4PT Ch. 3.1 - Prob. 5PT Ch. 3.1 - Prob. 6PT Ch. 3.2 - Prob. 1PT Ch. 3.2 - Prob. 2PT Ch. 3.2 - Prob. 3PT Ch. 3.2 - Prob. 4PT

Ch. 3.3 - Prob. 1PT Ch. 3.3 - Prob. 2PT Ch. 3.3 - Prob. 3PT Ch. 3.3 - Prob. 4PT Ch. 3.3 - Prob. 5PT Ch. 3.3 - Prob. 6PT Ch. 3.4 - Prob. 1PT Ch. 3.4 - Prob. 2PT Ch. 3.4 - Prob. 3PT Ch. 3.4 - Prob. 4PT Ch. 3.4 - Prob. 5PT Ch. 3.4 - Prob. 6PT Ch. 3.4 - Prob. 7PT Ch. 3.5 - Prob. 1PT Ch. 3.5 - Prob. 2PT Ch. 3.5 - Prob. 3PT Ch. 3.5 - Prob. 4PT Ch. 3.5 - Prob. 5PT Ch. 3.5 - Prob. 6PT Ch. 3.6 - Prob. 1PT Ch. 3.6 - Prob. 2PT Ch. 3.6 - Prob. 3PT Ch. 3.6 - Prob. 4PT Ch. 3.6 - Prob. 5PT Ch. 3.7 - Prob. 1PT Ch. 3.7 - Prob. 2PT Ch. 3.7 - Prob. 3PT Ch. 3.7 - Prob. 4PT Ch. 3.8 - Prob. 1PT Ch. 3.8 - A bacteria culture starts with 50 organisms and...Ch. 3.8 - Prob. 3PT Ch. 3.9 - A right triangle has one leg with constant length...Ch. 3.9 - Prob. 2PT Ch. 3.10 - Prob. 1PT Ch. 3.10 - Prob. 2PT Ch. 3.10 - Prob. 3PT Ch. 3.10 - Prob. 4PT Ch. 3.11 - Prob. 1PT Ch. 3.11 - Prob. 2PT Ch. 3.11 - Prob. 3PT Ch. 3.11 - Prob. 4PT Ch. 3.11 - Prob. 5PT

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Whether the statement “If y = f ( g ( x ) ) then y ′ = f ′ ( g ′ ( x ) ) ” is true or false.

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Whether the statement “If y=f(g(x)) then y′=f′(g′(x))” is true or false.

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