Answer
$$\frac{dr}{d\theta}=\theta^{-1/2}+\theta^{-1/3}+\theta^{-1/4}$$
Work Step by Step
$$r-2\sqrt\theta=\frac{3}{2}\theta^{2/3}+\frac{4}{3}\theta^{3/4}$$
$$r=2\sqrt\theta+\frac{3}{2}\theta^{2/3}+\frac{4}{3}\theta^{3/4}$$
$$r=2\theta^{1/2}+\frac{3}{2}\theta^{2/3}+\frac{4}{3}\theta^{3/4}$$
We do not have to use implicit differentiation here, but apply the derivative rules right away:
$$\frac{dr}{d\theta}=2\times\frac{1}{2}\theta^{-1/2}+\frac{3}{2}\times\frac{2}{3}\theta^{-1/3}+\frac{4}{3}\times\frac{3}{4}\theta^{-1/4}$$
$$\frac{dr}{d\theta}=\theta^{-1/2}+\theta^{-1/3}+\theta^{-1/4}$$
