# Power rule

(Redirected from Calculus with polynomials)

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

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 $n \ge 0$. It is considered the first general theorem of calculus to be discovered.

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

This is an indefinite integral where $C$ is the arbitrary constant of integration.

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

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

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

## 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.

For integers $n \geq 1$, the derivative of $f(x)=x^n \!$ is $f'(x)=nx^{n-1},\!$ that is,

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

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

$f'(a) = \lim_{x\rarr a} \frac{x^n-a^n}{x-a} = \lim_{x\rarr a} x^{n-1}+ax^{n-2}+ \cdots +a^{n-2}x+a^{n-1}$

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

$f'(a) = \lim_{x\rarr a} x^{n-1}+ax^{n-2}+ \cdots +a^{n-2}x+a^{n-1} = a^{n-1}+a^{n-1}+ \cdots +a^{n-1}+a^{n-1} = n\cdot a^{n-1}$

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:

$\left( \sum_{r=0}^n a_r x^r \right)' = \sum_{r=0}^n \left(a_r x^r\right)' = \sum_{r=0}^n a_r \left(x^r\right)' = \sum_{r=0}^n ra_rx^{r-1}.$

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

$\int\!\left( \sum^n_{k=0} a_k x^k\right)\,dx= \sum^n_{k=0} \frac{a_k x^{k+1}}{k+1} + C.$

## Generalizations

One can prove that the power rule is valid for any exponent r, that is

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

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

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

one can differentiate and integrate linear combinations of powers of 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.