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: 36


$\displaystyle \frac{dy}{dx}=\frac{-30x}{(3x^{2}-4)^{6}}$

Work Step by Step

$y=\displaystyle \frac{1}{(3x^{2}-4)^{5}}=(3x^{2}-4)^{-5}=[w(x)]^{-5}$ Use the chain rule. $\displaystyle \frac{dy}{dx}=\frac{d}{dx}[w(x)]^{-5}$ $\displaystyle \frac{dy}{dx}=-5(w(x))^{-6}\cdot w^{\prime}(x)$ $\displaystyle \frac{dy}{dx}=-5(3x^{2}-4)^{-6}\cdot 6x$ $\displaystyle \frac{dy}{dx}=-30x(3x^{2}-4)^{-6}$ $\displaystyle \frac{dy}{dx}=\frac{-30x}{(3x^{2}-4)^{6}}$
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