# How do you use the limit definition of the derivative to find the derivative of f(x)=4x^2?

Jul 16, 2016

The derivative of a function in a point $x$ is the:

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

#### Explanation:

Thus we evaluate:

${\lim}_{h \rightarrow 0} \frac{4 {\left(x + h\right)}^{2} - 4 {x}^{2}}{h} = {\lim}_{h \rightarrow 0} \frac{4 \left({x}^{2} + 2 x h + {h}^{2}\right) - 4 {x}^{2}}{h} = {\lim}_{h \rightarrow 0} \frac{4 {x}^{2} + 8 x h + 4 {h}^{2} - 4 {x}^{2}}{h} = {\lim}_{h \rightarrow 0} \frac{8 x h + 4 {h}^{2}}{h} = {\lim}_{h \rightarrow 0} \frac{h \left(8 x + 4 h\right)}{h} = {\lim}_{h \rightarrow 0} \left(8 x + 4 h\right) = 8 x$

So $\left(4 {x}^{2}\right) ' = 8 x$