Power Rule for Derivatives/Real Number Index/Proof 1

Proof
We are going to prove that $f' \left({x}\right) = n x^{n-1}$ 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 Exponential at Zero
 * $\displaystyle \lim_{x \to 0} \frac {\ln \left({1 + x}\right)} x = 1$ from Derivative of Logarithm at One

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

Hence the result.