Answer
$$\frac{dr}{d\theta}=-\sqrt{\frac{r}{\theta}}$$
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 $dr/d\theta$:
$$\frac{1}{2}r^{-1/2}\frac{dr}{d\theta}=-\frac{1}{2}\theta^{-1/2}$$
$$r^{-1/2}\frac{dr}{d\theta}=-\theta^{-1/2}$$
$$\frac{dr}{d\theta}=-\frac{\theta^{-1/2}}{r^{-1/2}}=-\frac{\sqrt r}{\sqrt\theta}=-\sqrt{\frac{r}{\theta}}$$
