Find the rate of change of the area of a circle with respect to its radius $r$ when

(a) $r = 3$ cm

(b) $r = 4$ cm

Let $A = \pi {r^2}$ …(1)
(where $A$ denotes the area of the circle when its radius is $r$)
Differentiating (1), w.r.t. $r$, we get
$= \cfrac{{dA}}{{dr}} = \pi (2r) = 2\pi r$

(a) ${\left( {\cfrac{{dA}}{{dr}}} \right)_{r = 3{\rm{cm}}}} = 2\pi (3){\rm{cm}} = 6\pi {\rm{c}}{{\rm{m}}^{\rm{2}}}{\rm{/cm}}$

(b) ${\left( {\cfrac{{dA}}{{dr}}} \right)_{r = 4{\rm{cm}}}} = 2\pi (4)cm = 8\pi {\rm{c}}{{\rm{m}}^{\rm{2}}}{\rm{/cm}}$