Derivative of Function to Power of Function

Theorem
Let $u \left({x}\right), v \left({x}\right)$ be real functions which are differentiable on $\R$.

Then:
 * $D_x \left({u^v}\right) = v u^{v-1} D_x \left({u}\right) + u^v \left({\ln u}\right) D_x \left({v}\right)$

Also see

 * Power Rule for Derivatives: when $u = x$ and $v = n$ where $n$ is constant:
 * $D_x \left({x^n}\right) = n x^{n-1}$


 * Derivative of Exponential Function: when $v = x$ and $u = a$ where $a$ is constant:
 * $D_x \left({a^x}\right) = a^x \ln a$