# How do you find the derivative of f(x)=x^3-12x using the limit process?

Jan 19, 2017

$\frac{\mathrm{df}}{\mathrm{dx}} = 3 {x}^{2} - 12$. Explanation below.

#### Explanation:

$f$ is a polynomial function, so it is (infinitely) differentiable everywhere. Using the alternate definition of the derivative, since we need the general derivative function:

${\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h} =$

${\lim}_{h \to 0} \frac{{\left(x + h\right)}^{3} - 12 \left(x + h\right) - {x}^{3} + 12 x}{h} =$

${\lim}_{h \to 0} \frac{{x}^{3} + 3 {x}^{2} h + 3 x {h}^{2} + {h}^{3} - 12 x - 12 h - {x}^{3} + 12 x}{h} =$

${\lim}_{h \to 0} \frac{3 {x}^{2} h + 3 x {h}^{2} + {h}^{3} - 12 h}{h} =$

${\lim}_{h \to 0} \frac{h \left(3 {x}^{2} + 3 x h + {h}^{2} - 12\right)}{h} =$

${\lim}_{h \to 0} \left(3 {x}^{2} + 3 x h + {h}^{2} - 12\right) = 3 {x}^{2} + 0 + 0 - 12$

$= 3 {x}^{2} - 12$.