derivative of xn

Recall the typical derivativeMathworldPlanetmathPlanetmath formulaMathworldPlanetmathPlanetmath is


The derivative of xn can be computed directly from this formula utilizing the binomial theorem, see for instance an alternative proof of the deriviative of xn (

However, to avoid invoking the binomial theorem one can often make use of alternative definitions of the derivative which are justified by inspecting a diagrams and/or through the use of algebra. Instead of using h, use h=x-a so that ax is the same as h0. This gives the formula


This is the standard slope formula between the two points (a,f(a)) and (x,f(x)) only now we let a approach x.

From this formula the casual rule


for positive integer values of n can be easily derived.

First notice that




When n is not a positive integer the proof typically depends on implicit differentiationMathworldPlanetmath as follows:


For the theoretically inclined this solution can be disappointing because it depends on a proof for the derivative of lnx. Most texts simply redefine lnx as the integralPlanetmathPlanetmath of 1/x or in some similar fashion delay an honest proof.

For this reason it is often instructive to prove the power ruleMathworldPlanetmathPlanetmath in stages depending on the type of exponents. Having proven the result for n a positive integer, one can extend this to -n using the product ruleMathworldPlanetmath.

To begin with observe 1=xnx-n. Therefore

0 = ddx(1)=ddx(xnx-n)=ddx(xn)x-n+xnddx(x-n)
= nxn-1x-n+xnddx(x-n)=nx+xnddx(x-n).

Now solve for ddx(x-n).


Likewise fractional powers can also be accommodated without the use of lnx by beginning with the property: given 1/b a rational numberPlanetmathPlanetmathPlanetmath then


Using the chain ruleMathworldPlanetmath we can prove the power rule for x1/b as follows.


Once again solve for ddx(x1/b)


Finally, the derivative of xa/b for any fraction a/b is done by observing that xa/b=(xa)1/b so indeed the chain rule once again solves the problem.

Title derivative of xn
Canonical name DerivativeOfXn
Date of creation 2013-03-22 15:50:10
Last modified on 2013-03-22 15:50:10
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 16
Author Algeboy (12884)
Entry type Theorem
Classification msc 26B05
Classification msc 26A24
Synonym Power rule
Related topic DerivativesByPureAlgebra
Related topic AlternativeProofOfDerivativeOfXn