Answer
$16 \pi $ or $\approx 50.265~ft^2/ft$
Work Step by Step
Consider the tangent line that contains the point $(x, c)$. The slope the tangent line to the graph of $f(x)$ at $(x, c)$ can be written as
$f'(x) =\lim\limits_{x \to c} \dfrac{f(x)-f(c)}{x-c}$
We use the above equation to find the rate of change of surface area with respect to the radius.
Consider $S(r)=4\pi r^2$. Then at $r=2$, we have:
$S'(2) =\lim\limits_{r \to 2} \dfrac{S(r)-S(2)}{r-2} \\=\lim\limits_{r \to 2} \dfrac{4\pi r^2-(4 \pi)(2^2)}{r-2}\\=\lim\limits_{r \to 2} \dfrac{4\pi (r^2-4) }{r-2}\\=\lim\limits_{r \to 2} \dfrac{ 4\pi (r-2)(r+2) }{r-2}\\=\lim\limits_{r \to 2} 4 \pi (r+2) \\=16 \pi \\ \approx 50.265~ft^2/ft$
