# Chapter 3 - Section 3.7 - Implicit Differentiation - Exercises - Page 164: 17

$$\frac{dr}{d\theta}=-\sqrt{\frac{r}{\theta}}$$

#### Work Step by Step

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

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