Power Rule for Derivatives/Real Number Index/Proof 1

Proof
We are going to prove that $\map {f'} x = n x^{n - 1}$ holds for all real $n$.

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

Now we use the following results:
 * $\displaystyle \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ from Derivative of Exponential at Zero
 * $\displaystyle \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$ from Derivative of Logarithm at One

to obtain:
 * $x^n \cdot \dfrac {e^{n \map \ln {1 + \frac h x} } - 1} {n \map \ln {1 + \dfrac h x} } \cdot \dfrac {n \map \ln {1 + \dfrac h x}} {\dfrac h x} \cdot \dfrac 1 x \to n x^{n - 1}$ as $h \to 0$

Hence the result.