University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.7 - Implicit Differentiation - Exercises - Page 164: 19



Work Step by Step

$$\sin(r\theta)=\frac{1}{2}$$ 1) Differentiate both sides of the equation with respect to $\theta$: $$\frac{d}{d\theta}\sin(r\theta)=\frac{d}{d\theta}\frac{1}{2}$$ $$\cos(r\theta)\frac{d}{d\theta}(r\theta)=0$$ $$\cos(r\theta)(r+\theta\frac{dr}{d\theta})=0$$ $$r\cos(r\theta)+\theta\cos(r\theta)\frac{dr}{d\theta}=0$$ 2) Collect all the terms with $dr/d\theta$ onto one side and solve for $dr/d\theta$: $$\theta\cos(r\theta)\frac{dr}{d\theta}=-r\cos(r\theta)$$ $$\frac{dr}{d\theta}=-\frac{r\cos(r\theta)}{\theta\cos(r\theta)}=-\frac{r}{\theta}$$
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