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: 4

Answer

$$\frac{dy}{dx}=-\csc^2 x\cos(\cot x)$$

Work Step by Step

$$y=\sin(\cot x)$$ $$\frac{dy}{dx}=\frac{d(\sin(\cot x))}{dx}$$ Let $u=\cot x$ and $y=\sin u$. Then, according to Chain Rule, $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$ $$\frac{dy}{dx}=\frac{d(\sin u)}{du}\frac{d(\cot x)}{dx}$$ $$\frac{dy}{dx}=\cos u\times(-\csc^2 x)$$ $$\frac{dy}{dx}=-\csc^2 x\cos(\cot x)$$
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