generalizations of the Leibniz rule

For the derivative, the product ruleMathworldPlanetmath


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


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


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


where i is a multi-index.


  • 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 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