PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (46)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Copyright
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
power rule (Theorem)

The power rule states that

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

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.

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


$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}x^0$ $\displaystyle =$ $\displaystyle \frac{0}{x} = 0 = \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}1$  
$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}x^1$ $\displaystyle =$ $\displaystyle 1x^0 = 1 = \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}x$  
$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}x^2$ $\displaystyle =$ $\displaystyle 2x$  
$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}x^3$ $\displaystyle =$ $\displaystyle 3x^2$  
$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}\sqrt{x}$ $\displaystyle =$ $\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}x^{1/2} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$  
$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}2x^e$ $\displaystyle =$ $\displaystyle 2ex^{e-1}$  



"power rule" is owned by mathcam. [ full author list (2) | owner history (1) ]
(view preamble | get metadata)

View style:

See Also: product rule, derivation, derivative


Attachments:
proof of the power rule (Proof) by mathcam
binomial proof of positive integer power rule (Proof) by mathcam
Log in to rate this entry.
(view current ratings)

Cross-references: formula, application, derivatives, sum rule, product rule, chain rule, Power rule

This is version 4 of power rule, born on 2002-02-24, modified 2003-07-30.
Object id is 2630, canonical name is PowerRule.
Accessed 5108 times total.

Classification:
AMS MSC: 26A03 (Real functions :: Functions of one variable :: Foundations: limits and generalizations, elementary topology of the line)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)