Power Rule for Derivatives/Real Number Index

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
We are going to prove that $f^{\prime}(x) = n x^{n-1}$ holds for all real $n$.

To do this, we compute the limit $\displaystyle \lim_{h \to 0} \frac{\left({x + h}\right)^n - x^n} h$:

Now we use the following results:
 * $\displaystyle \lim_{x \to 0} \frac {\exp x - 1} x = 1$ from Derivative of Exponent at Zero
 * $\displaystyle \lim_{x \to 0} \frac {\ln \left({1 + x}\right)} x = 1$ from Derivative of Logarithm at One

... to obtain:
 * $\displaystyle \frac {e^{n \ln \left({1 + \frac h x}\right)} - 1} {n \ln \left( {1 + \frac h x}\right)} \cdot \frac {n \ln \left({1 + \frac h x}\right)} {\frac h x} \cdot \frac 1 x \to n x^{n-1}$ as $h \to 0$

Hence the result.