Power Rule for Derivatives/Natural Number Index/Proof by Binomial Theorem

Proof
Let $f \left({x}\right) = x^n$ for $x \in \R, n \in \N$.

By the definition of the derivative:
 * $\displaystyle \frac {\mathrm d}{\mathrm d x} f \left({x}\right) = \lim_{h \to 0} \frac{f \left({x + h}\right) - f \left({x}\right)} h = \lim_{h \to 0} \frac{(x+h)^n - x^n} h$

Using the binomial theorem this simplifies to: