# power rule

 $\frac{\mathrm{d}}{\mathrm{d}x}x^{p}=px^{p-1},\quad p\in\mathbb{R}$

Repeated use of the above formula gives

 $\displaystyle\frac{d^{i}}{dx^{i}}x^{k}=\begin{cases}0&i>k\\ \frac{k!}{(k-i)!}x^{k-i}&i\leq k,\end{cases}$

for $i,k\in\mathbb{Z}$.

## Examples

 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}x^{0}$ $\displaystyle=$ $\displaystyle\frac{0}{x}=0=\frac{\mathrm{d}}{\mathrm{d}x}1$ $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}x^{1}$ $\displaystyle=$ $\displaystyle 1x^{0}=1=\frac{\mathrm{d}}{\mathrm{d}x}x$ $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}x^{2}$ $\displaystyle=$ $\displaystyle 2x$ $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}x^{3}$ $\displaystyle=$ $\displaystyle 3x^{2}$ $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\sqrt{x}$ $\displaystyle=$ $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}x^{1/2}=\frac{1}{2}x^{-1/2}=\frac{1% }{2\sqrt{x}}$ $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}2x^{e}$ $\displaystyle=$ $\displaystyle 2ex^{e-1}$
Title power rule PowerRule 2013-03-22 12:28:03 2013-03-22 12:28:03 mathcam (2727) mathcam (2727) 7 mathcam (2727) Theorem msc 26A03 ProductRule Derivation Derivative