Answer
$f = x^{3} $
$c = -2$
Work Step by Step
We have that $f'(c) = \lim\limits_{∆x \to 0}$ $\frac{(- 2 +∆x)^3 + 8 }{∆x}$.
Taken the original formula $f'(x) = \lim\limits_{∆x \to 0}$ $\frac{f(x + ∆x) - f(x)}{∆x}$, which is equal to
$f'(c) = \lim\limits_{∆x \to 0}$ $\frac{f(c + ∆x) - f(c)}{∆x}$, we have that
$f(c + ∆x) = (- 2 +∆x)^3$
Then, we can conclude that $f = x^{3} $ and $c = -2$
