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

Proof
Let $\map f x = x^n$ for $x \in \R, n \in \N$.

By the definition of the derivative:
 * $\ds \dfrac \d {\d x} \map f x = \lim_{h \mathop \to 0} \dfrac {\map f {x + h} - \map f x} h = \lim_{h \mathop \to 0} \dfrac {\paren {x + h}^n - x^n} h$

Using the binomial theorem this simplifies to: