Power function/Positive base/Real exponent/Fact

Let ${}\alpha \in \mathbb {R}$.

Then the

function

$f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} _{+},x\longmapsto x^{\alpha },$

is differentiable, and its derivative is

${}f'(x)=\alpha x^{\alpha -1}\,.$

Proof, Write another proof