Answer
see work for proof
Work Step by Step
The derivative function is defined as$\\\\$
$\displaystyle \frac{d}{dx}[f(x)]=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow 0}\frac{(x+h)^{5}-x^{5}}{h}\\\\$
$=\displaystyle \lim_{h\rightarrow 0}\frac{x^{5}+5x^{4}h+10x^{3}h^{2}+10x^{2}h^{3}+5xh^{4}+h^{5}-x^{5}}{h}\\\\$
$=\displaystyle \lim_{h\rightarrow 0}\frac{5x^{4}h+10x^{3}h^{2}+10x^{2}h^{3}+5xh^{4}+h^{5}}{h}\\\\$
$=\displaystyle \lim_{h\rightarrow 0}\frac{h(5x^{4}+10x^{3}h+10x^{2}h^{2}+5xh^{3}+h^{4})}{h}\\\\$
$=\displaystyle \lim_{h\rightarrow 0}(5x^{4}+10x^{3}h+10x^{2}h^{2}+5xh^{3}+h^{4})\\\\$
(all the terms approach 0, except the first)$\\\\$
$=5x^{4}$
So,$\\\\$
$\displaystyle \frac{d}{dx}[x^{5}]=5x^{4}$
