Answer
$$27~m^3/m$$
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 volume with respect to the side length.
Given $V(x)=x^3$, we have at $x=3$:
$V'(3) =\lim\limits_{x \to 3} \dfrac{V(x)-V(3)}{x-2} \\=\lim\limits_{x \to 3} \dfrac{x^3-3^3}{x-3} \\= \lim\limits_{x \to 3} \dfrac{x^3-27}{x-3} \\=\lim\limits_{x \to 3} \dfrac{(x-3) (x^2+3x+9)}{x-3}\\=\lim\limits_{x \to 3} x^2+3x+9 \\=27~m^3/m$
