Answer
$$\frac{dr}{d\theta}=-\frac{r}{\theta}$$
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 $dr/d\theta$:
$$\theta\cos(r\theta)\frac{dr}{d\theta}=-r\cos(r\theta)$$
$$\frac{dr}{d\theta}=-\frac{r\cos(r\theta)}{\theta\cos(r\theta)}=-\frac{r}{\theta}$$
