I'm looking for a proof by induction for the power rule of derivatives, i.e. $\displaystyle \frac{d}{dx}x^n=nx^{n-1}$ for $\displaystyle n\geq1$

In the proof, I need to use the following facts:

$\displaystyle

x^{n+1}=xx^n

$

and

$\displaystyle

g'(x)=x(f'(x))+f(x)

$

Can someone help out here?