Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.7 The Chain Rule - Exercises - Page 146: 28


\begin{align*} f'(g(x))&= \frac{-3}{(x+1)^4}\\ g'(f(x))&= \frac{-3x^2}{(x^3+1)^2} \end{align*}

Work Step by Step

Given $$f(u)=u^{3}, \quad u=g(x)=\frac{1}{x+1}$$ Since \begin{align*} f(g(x))&= [g(x)]^3\\ &= \frac{1}{(x+1)^3}\\ g(f(x))&= \frac{1}{f(x)+1}\\ &= \frac{1}{x^3+1} \end{align*} Then \begin{align*} f'(g(x))&= \frac{-3}{(x+1)^4}\\ g'(f(x))&= \frac{-3x^2}{(x^3+1)^2} \end{align*}
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