# power rule

The power rule states that

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

This rule, when combined with the chain rule, product rule, and sum rule, makes calculating many derivatives far more tractable. This rule can be derived by repeated application of the product rule. See the proof of the power rule (http://planetmath.org/ProofOfPowerRule).

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