Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 327: 33


$$ f'(\theta)= e^{\theta}\cos( e^{\theta}) .$$

Work Step by Step

Recall that $(e^x)'=e^x$ Recall that $(\sin x)'=\cos x$. Since we have $$ f(\theta)= \sin(e^{\theta})$$ then the derivative $ f'(\theta)$, using the chain rule, is given by $$ f'(\theta)=\cos( e^{\theta}) (e^{\theta})'=e^{\theta}\cos( e^{\theta}) .$$
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