How to proof $(x^n)' = nx^{n-1} (with \enspace n \in \mathbb{R})$?

Question

It's trivial to proove that :

$$(x^n)' = nx^{n-1} \enspace (with \enspace n \in \mathbb{N}) .$$

But how to proove it with $n \in \mathbb{R}$ ?

Because it works.

Example :

$(x^{\frac{1}{2}}) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$

...etc.

Thank you for this help.