# Power Rule for Derivatives/Corollary

## Corollary to Power Rule for Derivatives

Let $n \in \R$.

Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.

Then:

$\map {\dfrac \d {\d x} } {c x^n} = n c x^{n - 1}$

everywhere that $\map f x = x^n$ is defined.

## Proof

 $\displaystyle \map {\frac \d {\d x} } {c x^n}$ $=$ $\displaystyle c \, \map {\frac \d {\d x} } {x^n}$ Derivative of Constant Multiple $\displaystyle$ $=$ $\displaystyle n c x^{n - 1}$ Power Rule for Derivatives

$\blacksquare$