Question:
Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 cm?
Solution:
Let V be the volume of the spherical ball. Then,
$V=\frac{4}{3} \pi r^{3}$
$\Rightarrow \frac{d V}{d r}=4 \pi r^{2}$'
Thus, the rate of change of the volume of the sphere is $4 \pi r^{2}$.
When $r=2 \mathrm{~cm}$
$\left(\frac{d V}{d r}\right)_{r=2}=4 \pi(2)^{2}$
$=16 \pi \mathrm{cm}^{3} / \mathrm{cm}$