Derivative of x to the a x

Theorem
Let $x \in \R$ be a real variable whose domain is the set of (strictly) positive real numbers $\R_{>0}$.

Let $c \in \R_{>0}$ be a (strictly) positive real constant.

Then:
 * $\dfrac \d {\d x} x^{a x} = a x^{a x} \paren {\ln x + 1}$

Proof
Let $y := x^{a x}$.

As $x > 0$, we can take the natural logarithm of both sides: