Power function/Positive base/Real exponent/Fact/Proof

Proof

By definition, we have

${}x^{\alpha }=\exp {\left(\alpha \,\ln x\right)}\,.$

The derivative with respect to ${}x$ equals

${}{\left(x^{\alpha }\right)}'={\left(\exp {\left(\alpha \,\ln x\right)}\right)}'={\frac {\alpha }{x}}\cdot \exp {\left(\alpha \,\ln x\right)}={\frac {\alpha }{x}}x^{\alpha }=\alpha x^{\alpha -1}\,$

using fact, fact and the chain rule.

To fact