Derivative of x to the x

Theorem

 * $\dfrac {\mathrm d} {\mathrm d x} x^x = x^x \left({\ln x + 1}\right)$

for $x > 0$.

Proof
Note that the Power Rule cannot be used because the index is not a constant.

Define $y$ as $x^x$.

As $x$ was stipulated to be positive, we can take the natural logarithm of both sides: