Answer
See below.
Work Step by Step
We can rewrite any real power of x as a power of $e$, like so:
$x^n= e^{n\ln x}$
with the restriction $x\gt0$
On taking the derivative, we get:
$\frac{d(x^n)}{dx}=\frac{d(e^{n\ln x})}{dx}$
Applying the chain rule, gives us:
$\frac{d(x^n)}{dx}=(e^{n\ln x})\frac{d({n\ln x})}{dx}$
$\frac{d(x^n)}{dx}=(e^{n\ln x})\frac{({n})}{x}$
$\frac{d(x^n)}{dx}=(x^n)\frac{({n})}{x}$
$\frac{d(x^n)}{dx}=nx^{(n-1)}$
Which is the power rule for derivatives.
