# Power rule

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In calculus, the power rule is one of the most important differentiation rules. Since differentiation is linear, polynomials can be differentiated using this rule.

${\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1},\qquad n\neq 0.}$

The power rule holds for all powers except for the constant value ${\displaystyle x^{0}}$ which is covered by the constant rule. The derivative is just ${\displaystyle 0}$ rather than ${\displaystyle 0\cdot x^{-1}}$ which is undefined when ${\displaystyle x=0}$.

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

${\displaystyle \int \!x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C,\qquad n\neq -1.}$

The integration of ${\displaystyle x^{-1}}$ requires a separate rule.

${\displaystyle \int \!x^{-1}\,dx=\ln |x|+C,}$

## Power rule

Historically the power rule was derived as the inverse of Cavalieri's quadrature formula which gave the area under ${\displaystyle x^{n}}$ for any integer ${\displaystyle n\geq 0}$. Nowadays the power rule is derived first and integration considered as its inverse.

${\displaystyle \left(x^{n}\right)'=nx^{n-1}.}$

The power rule for integration

${\displaystyle \int \!x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C}$

for ${\displaystyle n\geq 0}$ is then an easy consequence. One just needs to take the derivative of this equality and use the power rule and linearity of differentiation on the right-hand side.

### Proof

To prove the power rule for differentiation, we use the definition of the derivative as a limit. But first, note the factorization for ${\displaystyle n\geq 1}$:

${\displaystyle f(x)-f(a)=x^{n}-a^{n}=(x-a)(x^{n-1}+ax^{n-2}+\cdots +a^{n-2}x+a^{n-1})}$

Using this, we can see that

${\displaystyle f'(a)=\lim _{x\rightarrow a}{\frac {x^{n}-a^{n}}{x-a}}=\lim _{x\rightarrow a}x^{n-1}+ax^{n-2}+\cdots +a^{n-2}x+a^{n-1}}$

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

${\displaystyle f'(a)=\lim _{x\rightarrow a}x^{n-1}+ax^{n-2}+\cdots +a^{n-2}x+a^{n-1}=a^{n-1}+a^{n-1}+\cdots +a^{n-1}+a^{n-1}=n\cdot a^{n-1}}$

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 ${\displaystyle n}$ . For an irrational ${\displaystyle n}$, a rational approximation is appropriate.

## Differentiation of arbitrary polynomials

To differentiate arbitrary polynomials, one can use the linearity property of the differential operator to obtain:

${\displaystyle \left(\sum _{r=0}^{n}a_{r}x^{r}\right)'=\sum _{r=0}^{n}\left(a_{r}x^{r}\right)'=\sum _{r=0}^{n}a_{r}\left(x^{r}\right)'=\sum _{r=0}^{n}ra_{r}x^{r-1}.}$

Using the linearity of integration and the power rule for integration, one shows in the same way that

${\displaystyle \int \!\left(\sum _{k=0}^{n}a_{k}x^{k}\right)\,dx=\sum _{k=0}^{n}{\frac {a_{k}x^{k+1}}{k+1}}+C.}$

## Generalizations

One can prove that the power rule is valid for any exponent Template:Mvar, that is

${\displaystyle \left(x^{r}\right)'=rx^{r-1},}$

as long as Template:Mvar is in the domain of the functions on the left and right hand sides and Template:Mvar is nonzero. Using this formula, together with

${\displaystyle \int \!x^{-1}\,dx=\ln |x|+C,}$

one can differentiate and integrate linear combinations of powers of Template:Mvar 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.