Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.4 - The Chain Rule - 3.4 Exercises - Page 204: 3


$$\frac{dy}{dx}=\pi\sec^2({\pi x})$$

Work Step by Step

$$y=\tan\pi x$$ $$\frac{dy}{dx}=\frac{d(\tan\pi x)}{dx}$$ Let $u=\pi x$ and $y=\tan u$. Then, according to Chain Rule, $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$ $$\frac{dy}{dx}=\frac{d(\tan u)}{du}\frac{d(\pi x)}{dx}$$ $$\frac{dy}{dx}=\sec^2 u\times\pi$$ $$\frac{dy}{dx}=\pi\sec^2({\pi x})$$
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