Answer
$16 \pi \text{ or} \approx 50.265\text{ ft}^2 \text{ per foot}$
Work Step by Step
Let us say that the tangent Line contains the points $(x, c)$. The slope the tangent Line to the graph of $f(x)$ at $(x, c)$ can be written as
$\\$
$$\\f'(x) =\lim\limits_{x \to c} \dfrac{f(x)-f(c)}{x-c} \text{ ... (1)}$$
In order to find the rate of change of surface area with respect to radius, we will use equation (1).
Plug in the given data to obtain:
$$S'(2) =\lim\limits_{r \to 2} \dfrac{S(r)-S(2)}{r-2} \\=\lim\limits_{r \to 2} \dfrac{4\pi r^2-(4 \pi) \times ((2)^2)}{(r-2)}\\=\lim\limits_{r \to 2} \dfrac{4\pi (r^2-4) }{(r-2)}\\=\lim\limits_{r \to 2} \dfrac{ 4\pi (r-2)(r+2) }{(r-2)}\\=\lim\limits_{r \to 2} 4 \pi \times (r+2) \\=16 \pi \\ \approx 50.265 \ ft^2/ft$$
