Derivative of Function to Power of Function

Theorem
Let $\map u x, \map v x$ be real functions which are differentiable on $\R$.

Then:
 * $\map {\dfrac \d {\d x} } {u^v} = v u^{v - 1} \map {\dfrac \d {\d x} } u + u^v \paren {\ln u} \map {\dfrac \d {\d x} } v$

Also see

 * Power Rule for Derivatives: when $u = x$ and $v = n$ where $n$ is constant:
 * $\map {\dfrac \d {\d x} } {x^n} = n x^{n-1}$


 * Derivative of Exponential Function: when $v = x$ and $u = a$ where $a$ is constant:
 * $\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$