Power Rule for Derivatives

Theorem
Let $n \in \R$.

Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = x^n$.

Then:
 * $f^{\prime} \left({x}\right) = n x^{n-1}$

everywhere that $f \left({x}\right) = x^n$ is defined.

When $x = 0$ and $n = 0$, $f^{\prime} \left({x}\right)$ is undefined.

Corollary

 * $\displaystyle \frac {\mathrm d}{\mathrm d x} \left({c x^n}\right) = n c x^{n-1}$

Proof
This can be done in sections.

Proof of Corollary
Follows directly from the above, and Derivative of Constant Multiple.