Power rule

Topics in Calculus

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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.

$\frac{d}{dx} x^n = nx^{n-1}.$

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 ≥ 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.

Hence, the derivative of x¹⁰⁰ is 100x⁹⁹ and the indefinite integral of x¹⁰⁰ is 1/101 x ¹⁰¹ +C where C is an arbitrary constant of integration.

Power rule

The power rule is for polynomials and states that for every integer n, 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 natural n 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 case of $n = 0$ is trivial because $x^0=1$, so $\frac{d}{dx}1=0=0\cdot x^{-1}$.

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.