Power rule

In mathematics, the power rule is one of the most important differentiation rules in calculus. 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

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

Generalization

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.

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