## University Calculus: Early Transcendentals (3rd Edition)

The rate of change of the area $A$ of the circle with respect to $r=3$ is $6\pi$.
The function of area of the circle with a radius $r$ is $f(r)=\pi r^2$. The rate of change of the area $A$ of the circle with respect to $r=3$ is the derivative $f'(3)$, which can be calculated by $$f'(3)=\lim_{h\to0}\frac{f(h+3)-f(3)}{h}=\lim_{h\to0}\frac{\pi(h+3)^2-3^2\pi}{h}$$ $$f'(3)=\lim_{h\to0}\frac{\pi(h^2+6h+9)-9\pi}{h}=\lim_{h\to0}\frac{\pi h^2+6\pi h}{h}=\lim_{h\to0}(\pi h+6\pi)$$ $$f'(3)=\pi\times0+6\pi=6\pi$$ The rate of change of the area $A$ of the circle with respect to $r=3$ is $6\pi$.