Faà di Bruno’s formula


Faà di Bruno’s formulaMathworldPlanetmathPlanetmath is a generalizationPlanetmathPlanetmath of the chain ruleMathworldPlanetmath to higher order derivatives which expresses the derivativePlanetmathPlanetmath of a compositionMathworldPlanetmathPlanetmath of functions as a series of productsPlanetmathPlanetmath of derivatives:

dndxnf(g(x))=k=0nkmk=nn!m1!m2!m3!1!m1 2!m2 3!m3f(m1++mn)(g(x))j:mj0(g(j)(x))mj

This formula was discovered by Francesco Faà di Bruno in the 1850s and can be proved by inductionMathworldPlanetmath on the order of the derivative.

References

  • 1 Faà di Bruno, C. F.. “Sullo sviluppo delle funzione.” Ann. di Scienze Matem. et Fisiche di Tortoloni 6 (1855): 479-480
  • 2 Faà di Bruno, C. F.. “Note sur un nouvelle formule de calcul différentiel.” Quart. J. Math. 1 (1857): 359-360
  • 3 H. Figueroa & J. M. Gracia-Bondía, “Combinatorial Hopf Algebras in Quantum Field Theory I” Rev. Math. Phys. 17 (2005): 881 - 975
Title Faà di Bruno’s formula
Canonical name FaaDiBrunosFormula
Date of creation 2013-03-22 16:38:57
Last modified on 2013-03-22 16:38:57
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Definition
Classification msc 16W30
Synonym Faa di Bruno’s formula
Synonym Faà di Bruno formula
Synonym Faa di Bruno formula