binomial proof of positive integer power rule

We will use the difference quotient in this proof of the power ruleMathworldPlanetmathPlanetmath for positive integers. Let f(x)=xn for some integer n0. Then we have


We can use the binomial theoremMathworldPlanetmath to expand the numerator


where Ckn=n!k!(n-k)!. We can now simplify the above

f(x) =limh0hn+nxhn-1++nxn-1h+xn-xnh
Title binomial proof of positive integer power rule
Canonical name BinomialProofOfPositiveIntegerPowerRule
Date of creation 2013-03-22 12:29:43
Last modified on 2013-03-22 12:29:43
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Proof
Classification msc 26A03