$ \frac{d}{d x} x^{n}=n x^{n-1} $ #### Proof of the Positive Integer Power Rule The formula $ z^{n}-x^{n}=(z-x)\left(z^{n-1}+z^{n-2} x+\cdots+z x^{n-2}+x^{n-1}\right) $ can be verified by multiplying out the right-hand side. Then from the alternative formula for the definition of the derivative, $ \begin{aligned} f^{\prime}(x) &=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}=\lim _{z \rightarrow x} \frac{z^{n}-x^{n}}{z-x} \\ &=\lim _{z \rightarrow x}\left(z^{n-1}+z^{n-2} x+\cdots+z x^{n-2}+x^{n-1}\right) \quad n \text { terms } \\ &=n x^{n-1} \end{aligned} $