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: 50


$$ y'= -\cos x\sin(\sin x)\cos(\cos(\sin x)).$$

Work Step by Step

Since $ y=\sin(\cos(\sin x))$, then by using the chain rule: $(f(g(x)))^{\prime}=f^{\prime}(g(x)) g^{\prime}(x)$ and using that $(\sin x)'=\cos x $ and $(\cos x)'=-\sin x $, the derivative $ y'$ is given by $$ y'=\cos(\cos(\sin x))(-\sin(\sin x)(\cos x))\\=-\cos x\sin(\sin x)\cos(\cos(\sin x)).$$
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