Real exponential function/Base/Derivative/Fact/Proof

Proof

By definition, we have

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

The derivative with respect to ${}x$ equals

${}{\left(a^{x}\right)}'={\left(\exp {\left(x\,\ln a\right)}\right)}'={\left(\ln a\right)}\exp '(x\,\ln a)={\left(\ln a\right)}\exp {\left(x\,\ln a\right)}={\left(\ln a\right)}a^{x}\,,$

due to fact and the chain rule.

To fact