Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm.
Question:

Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm.

Solution:

Let A be the area of the circular disc. Then,

$A=\pi r^{2}$

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

Let C be the circumference of the circular disc. Then,

$C=2 \pi r$

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

$\therefore \frac{d A}{d C}=\frac{d A / d r}{d C / d r}$

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

$\Rightarrow\left(\frac{d A}{d C}\right)_{r=3}=3 \mathrm{~cm}$