## Calculus: Early Transcendentals (2nd Edition)

The value of the derivative is $$\left.\frac{dA}{dr}\right|_{r=3}=6\pi.$$
Using the definition of the derivative with $r=3$ we have $$\left.\frac{dA}{dr}\right|_{r=3}=\lim_{h\to0}\frac{A(3+h)-A(3)}{h}=\lim_{h\to0}\frac{\pi(3+h)^2-\pi\cdot3^2}{h}=\lim_{h\to0}\frac{\pi(9+h^2+6h)-9\pi}{h}=\lim_{h\to0}\frac{9\pi+\pi h^2+6\pi h-9\pi}{h}=\lim_{h\to0}\frac{\pi h^2+6\pi h}{h}=\lim_{h\to0}(\pi h+6\pi)=\pi\cdot 0+6\pi=6\pi.$$