Answer
see work for proof.
Work Step by Step
The derivative function is defined as$\\$
$\displaystyle \frac{d}{dx}[f(x)]=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow 0}\frac{(x+h)^{4}-x^{4}}{h}\\\\$
$=\displaystyle \lim_{h\rightarrow 0}\frac{x^{4}+4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}-x^{4}}{h}\\\\$
$=\displaystyle \lim_{h\rightarrow 0}\frac{4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}}{h}\\\\$
$=\displaystyle \lim_{h\rightarrow 0}\frac{h(4x^{3}+6x^{2}h+4xh^{2}+h^{3})}{h}\\\\$
$=\displaystyle \lim_{h\rightarrow 0}(4x^{3}+6x^{2}h+4xh^{2}+h^{3})=4x^{3}\\\\$
(all other terms approach 0)$\\$
So,$\\\\$
$\displaystyle \frac{d}{dx}[x^{4}]=4x^{3}$
