generalizations of the Leibniz rule


For the derivative, the product ruleMathworldPlanetmath

(fg)=fg+fg

is known as the Leibniz rulePlanetmathPlanetmath. Below are various ways it can be generalized.

Higher derivatives

Let f,g be real (or complex) functionsMathworldPlanetmath defined on an open intervalDlmfPlanetmath of . If f and g are k times differentiableMathworldPlanetmathPlanetmath, then

(fg)(k)=r=0k(kr)f(k-r)g(r).

Generalized Leibniz rule for more functions

Let f1,,fr be real (or complex) valued functions that are defined on an open interval of . If f1,,fr are n times differentiable, then

dndtni=1rfi(t)=n1++nr=n(nn1,n2,,nr)i=1rdnidtnifi(t).

where (nn1,n2,,nr) is the multinomial coefficientDlmfMathworldPlanetmath.

Leibniz rule for multi-indices

If f,g:n are smooth functions defined on an open set of n, and j is a multi-index, then

j(fg)=ij(ji)i(f)j-i(g),

where i is a multi-index.

References

  • 1 Leibniz, Gottfried W. Symbolismus memorabilis calculi Algebraici et Infinitesimalis, in comparatione potentiarum et differentiarum; et de Lege Homogeneorum Transcendentali, Miscellanea Berolinensia ad incrementum scientiarum, ex scriptis Societati Regiae scientarum pp. 160-165 (1710). Available online at the http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=01-misc/1&seite:int=184digital library of the Berlin-Brandenburg Academy.
Title generalizations of the Leibniz rule
Canonical name GeneralizationsOfTheLeibnizRule
Date of creation 2013-03-22 14:30:18
Last modified on 2013-03-22 14:30:18
Owner GeraW (6138)
Last modified by GeraW (6138)
Numerical id 13
Author GeraW (6138)
Entry type Theorem
Classification msc 26A06
Synonym Leibniz rule
Related topic MultinomialTheorem
Related topic NthDerivativeOfADeterminant