Answer
$\displaystyle \frac{dV}{dt}=4\pi r^{2}\cdot\frac{dr}{dt}$
Work Step by Step
$r=r(t)$ so, by the chain rule,
$\displaystyle \frac{dV}{dt}=\frac{dV}{dr}\frac{dr}{dt}$
$\displaystyle \frac{ds}{dr}=\frac{d}{dr}[\frac{4\pi}{3}\cdot r^{3}]=\frac{4\pi}{3}\cdot 3r^{2}=4\pi r^{2}$
$\displaystyle \frac{dV}{dt}=4\pi r^{2}\cdot\frac{dr}{dt}$
