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
Canonical name	PowerRule
Date of creation	2013-03-22 12:28:03
Last modified on	2013-03-22 12:28:03
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 26A03
Related topic	ProductRule
Related topic	Derivation
Related topic	Derivative