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{{Calculus |Differential}}

In [[calculus]], the '''power rule''' is one of the most important [[differentiation rules]]. Since differentiation is linear, [[polynomial]]s can be differentiated using this rule.

:<math> \frac{d}{dx} x^n = nx^{n-1} , \qquad n \neq 0.</math>
The power rule holds for all powers except for the constant value <math>x^0</math> which is covered by the constant rule. The derivative is just <math>0</math> rather than <math>0 \cdot x^{-1}</math> which is undefined when <math>x=0</math>.

The inverse of the power rule enables all powers of a variable <math>x</math> except <math>x^{-1}</math> to be integrated. This integral is called [[Cavalieri's quadrature formula]] and was first found in a geometric form by [[Bonaventura Cavalieri]] for <math>n \ge 0</math>. It is considered the first general theorem of calculus to be discovered.

:<math>\int\! x^n \,dx= \frac{ x^{n+1}}{n+1} + C, \qquad n \neq -1.</math>

This is an [[indefinite integral]] where <math>C</math> is the [[arbitrary constant of integration]].

The integration of <math>x^{-1}</math> requires a separate rule.

:<math>\int \! x^{-1}\, dx= \ln |x|+C,</math>

Hence, the derivative of <math>x^{100}</math> is <math>100 x^{99}</math> and the integral of <math>x^{100}</math> is <math> \frac{1}{101} x^{101} +C</math>.

== Power rule==
Historically the power rule was derived as the inverse of [[Cavalieri's quadrature formula]] which gave the area under <math>x^n</math> for any integer <math>n \geq 0</math>. Nowadays the power rule is derived first and integration considered as its inverse.

For integers <math>n \geq 1</math>, the derivative of <math>f(x)=x^n \!</math> is <math>f'(x)=nx^{n-1},\!</math> that is,
:<math>\left(x^n\right)'=nx^{n-1}.</math>

The power rule ''for integration''
:<math>\int\! x^n \, dx=\frac{x^{n+1}}{n+1}+C</math>
for <math>n \geq 0</math> is then an easy consequence. One just needs to take the derivative of this equality and use the power rule and [[linear transformation|linearity]] of differentiation on the right-hand side.

===Proof===

To prove the power rule for differentiation, we use the [[derivative#Definition via difference quotients|definition of the derivative]] as a [[Limit of a function|limit]]. But first, note the [[Factorization of polynomials|factorization]] for <math>n \geq 1</math>:

:<math>f(x)-f(a) = x^n-a^n = (x-a)(x^{n-1}+ax^{n-2}+ \cdots +a^{n-2}x+a^{n-1})</math>

Using this, we can see that

:<math>f'(a) = \lim_{x\rarr a} \frac{x^n-a^n}{x-a} = \lim_{x\rarr a} x^{n-1}+ax^{n-2}+ \cdots +a^{n-2}x+a^{n-1} </math>

Since the division has been eliminated and we have a continuous function, we can freely substitute to find the limit:

:<math>f'(a) = \lim_{x\rarr a} x^{n-1}+ax^{n-2}+ \cdots +a^{n-2}x+a^{n-1} = a^{n-1}+a^{n-1}+ \cdots +a^{n-1}+a^{n-1} = n\cdot a^{n-1} </math>

The use of the quotient rule allows the extension of this rule for n as a negative integer, and the use of the laws of exponents and the [[chain rule]] allows this rule to be extended to all rational values of <math>n</math> . For an irrational <math>n</math>, a rational approximation is appropriate.

==Differentiation of arbitrary polynomials==

To differentiate arbitrary polynomials, one can use the [[linear transformation|linearity property]] of the [[differential operator]] to obtain:

:<math>\left( \sum_{r=0}^n a_r x^r \right)' =
\sum_{r=0}^n \left(a_r x^r\right)' =
\sum_{r=0}^n a_r \left(x^r\right)' =
\sum_{r=0}^n ra_rx^{r-1}.</math>

Using the linearity of integration and the power rule for integration, one shows in the same way that
:<math>\int\!\left( \sum^n_{k=0} a_k x^k\right)\,dx= \sum^n_{k=0} \frac{a_k x^{k+1}}{k+1} + C.</math>

==Generalizations==

One can prove that the power rule is valid for any exponent {{mvar|r}}, that is

:<math>\left(x^r\right)' = rx^{r-1},</math>

as long as {{mvar|x}} is in the domain of the functions on the left and right hand sides and {{mvar|r}} is nonzero. Using this formula, together with
:<math>\int \! x^{-1}\, dx= \ln |x|+C,</math>
one can differentiate and integrate linear combinations of powers of {{mvar|x}} which are not necessarily polynomials.

==References==
* Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). ''Calculus of a Single Variable: Early Transcendental Functions'' (3rd edition). Houghton Mifflin Company. ISBN 0-618-22307-X.

[[Category:Calculus]]
[[Category:Articles containing proofs]]

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