ff is a polynomial function, so it is (infinitely) differentiable everywhere. Using the alternate definition of the derivative, since we need the general derivative function:
lim_(h->0) (f(x + h) - f(x))/h =
lim_(h->0) ((x+h)^3 - 12(x + h) - x^3 + 12x)/h =
lim_(h->0) (x^3 + 3x^2h + 3xh^2 + h^3 - 12x - 12h - x^3 + 12x)/h =
lim_(h->0) (3x^2h + 3xh^2 + h^3 - 12h)/h =
lim_(h->0) (h(3x^2 + 3xh + h^2 - 12))/h =
lim_(h->0) (3x^2 + 3xh + h^2 - 12) = 3x^2 + 0 + 0 - 12
=3x^2 - 12.
