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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) ]
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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
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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 5198 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
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