Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm.
Question:

Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm.

Solution:

Let A be area of the circle. Then,

$A=\pi r^{2}$

$\Rightarrow \frac{d A}{d r}=2 \pi r$

Hence, the rate of change of the area of the circle is 2πr2πr.
When r = 5 cm,

$\left(\frac{d A}{d r}\right)_{r=5}=2 \pi(5)$

$=10 \pi \mathrm{cm}^{2} / \mathrm{cm}$