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.

Proof
This can be done in sections.

Historical Note
This result was obtained by in 1676. It was stated, without proof or explanation, by in his 1684 article Nova methodus pro maximis et minimis.

However, in a privately circulated paper of 1669, established exactly the same result, by investigation the nature of a function whose area under the graph is $x^m$.