Answer
$f'(x)=3x^2-12$
Work Step by Step
To take the derivative of a function using the limit process, plug into the equation $f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$ and simplify:
$f'(x)=\lim\limits_{h \to 0}\frac{[(x+h)^3-12(x+h)]-(x^3-12x)}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{x^3+2x^2h+xh^2+hx^2+2xh^2+h^3-12x-12h-x^3+12x}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{2x^2h+xh^2+hx^2+2xh^2+h^3-12h}{h}$
$f'(x)=\lim\limits_{h \to 0}2x^2+xh+x^2+2xh+h^2-12$
When you can't simplify any further, plug $0$ in for $h$ and simplify:
$f'(x)=2x^2+x(0)+x^2+2x(0)+(0)^2-12=2x^2+x^2-12=3x^2-12$
