# 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.

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

$\int \!x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C,\qquad n\neq -1.$ The integration of $x^{-1}$ requires a separate rule.

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

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

$\left(x^{n}\right)'=nx^{n-1}.$ The power rule for integration

$\int \!x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C$ for $n\geq 0$ 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 $n\geq 1$ :

$f(x)-f(a)=x^{n}-a^{n}=(x-a)(x^{n-1}+ax^{n-2}+\cdots +a^{n-2}x+a^{n-1})$ Using this, we can see that

$f'(a)=\lim _{x\rightarrow a}{\frac {x^{n}-a^{n}}{x-a}}=\lim _{x\rightarrow a}x^{n-1}+ax^{n-2}+\cdots +a^{n-2}x+a^{n-1}$ Since the division has been eliminated and we have a continuous function, we can freely substitute to find the limit:

$f'(a)=\lim _{x\rightarrow a}x^{n-1}+ax^{n-2}+\cdots +a^{n-2}x+a^{n-1}=a^{n-1}+a^{n-1}+\cdots +a^{n-1}+a^{n-1}=n\cdot a^{n-1}$ 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 $n$ . For an irrational $n$ , 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:

$\left(\sum _{r=0}^{n}a_{r}x^{r}\right)'=\sum _{r=0}^{n}\left(a_{r}x^{r}\right)'=\sum _{r=0}^{n}a_{r}\left(x^{r}\right)'=\sum _{r=0}^{n}ra_{r}x^{r-1}.$ Using the linearity of integration and the power rule for integration, one shows in the same way that

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

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

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

$\int \!x^{-1}\,dx=\ln |x|+C,$ one can differentiate and integrate linear combinations of powers of Template:Mvar which are not necessarily polynomials.