proof of generalized Leibniz rule

The generalized Leibniz rulePlanetmathPlanetmath can be derived from the plain Leibniz rule by inductionMathworldPlanetmath on r.

If r=2, the generalized Leibniz rule reduces to the plain Leibniz rule. This will be the starting point for the induction. To completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the induction, assume that the generalized Leibniz rule holds for a certain value of r; we shall now show that it holds for r+1.

Write i=1r+1fi(t)=(fr+1(t))(i=1r+1fi(t)). Applying the plain Leibniz rule,


By the generalized Leibniz rule for r (assumed to be true as the induction hypothesis), this equals


Note that


This is an immediate consequence of the expression for multinomial coefficientsMathworldPlanetmath as quotients of factorialsMathworldPlanetmath. Using this identity, the quantity can be written as


which is the generalized Leibniz rule for the case of r+1.

Title proof of generalized Leibniz rule
Canonical name ProofOfGeneralizedLeibnizRule
Date of creation 2013-03-22 14:34:14
Last modified on 2013-03-22 14:34:14
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Proof
Classification msc 26A06