Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 4 - Calculating the Derivative - 4.3 The Chain Rule - 4.3 Exercises - Page 225: 44


$a.\quad-\displaystyle \frac{18}{7}$ $b.\quad-5$

Work Step by Step

$D_{x}(f[g(x)]) =f^{\prime}[g(x)]\cdot g^{\prime}(x)$ (a) $D_{x}(g[f(x)])$ at $x=1$ $=g^{\prime}[f(1)]\cdot f^{\prime}(1)$ ... read the table: $f(1)=2,\quad f^{\prime}(1)=-6$ $=g^{\prime}(2)\cdot(-6)$ ... read the table:$ g^{\prime}(2)=\displaystyle \frac{3}{7}$ $=\displaystyle \frac{3}{7}(-6)$ $=\displaystyle \color{red}{-\frac{18}{7}}$ (b) $D_{x}(g[f(x)])$ at $x=2$ $=g^{\prime}[f(2)]\cdot f^{\prime}(2)$ ... read the table: $f(2)=4,\quad f^{\prime}(2)=-7$ $=g^{\prime}(4)\cdot(-7)$ ... read the table: $g^{\prime}(4)\displaystyle \cdot\frac{5}{7}$ $=\displaystyle \frac{5}{7}(-7)$ $=\color{red}{-5}$
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