Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises: 26

Answer

$$\frac{\partial}{\partial r}\sin(r\cos\theta)=\cos\theta\cos(r\cos\theta).$$ $$\frac{\partial}{\partial \theta}\sin(r\cos\theta)=-r\sin\theta\cos(r\cos\theta).$$

Work Step by Step

Regard $\theta$ as a constant and differentiate with respect to $r$: $$\frac{\partial}{\partial r}\sin(r\cos\theta)=\cos(r\cos\theta)\frac{\partial}{\partial r}(r\cos\theta)=\cos(r\cos\theta)\cos\theta\frac{\partial}{\partial r}r=\cos\theta\cos(r\cos\theta).$$ Regard $r$ as a constant and differentiate with respect to $\theta$: $$\frac{\partial}{\partial \theta}\sin(r\cos\theta)=\cos(r\cos\theta)\frac{\partial}{\partial \theta}(r\cos\theta)=\cos(r\cos\theta)r\frac{\partial}{\partial\theta}(\cos\theta)=-r\sin\theta\cos(r\cos\theta).$$
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