TOPICS
Search

Faà di Bruno's Formula

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

Faà di Bruno's formula gives an explicit equation for the nth derivative of the composition f(g(t)). If f(t) and g(t) are functions for which all necessary derivatives are defined, then

D^nf(g(t))=sum(n!)/(k_1!...k_n!)(D^kf)(g(t))((Dg(t))/(1!))^(k_1)...((D^ng(t))/(n!))^(k_n),
(1)

where k=k_1+...+k_n and the sum is over all partitions of n, i.e., values of k_1, ..., k_n such that

k_1+2k_2+...+nk_n=n
(2)

(Roman 1980).

It can also be expressed in terms of Bell polynomial B_(n,k)(x) as

D^nf(g(t))=sum_(k=0)^n(D^kf)(g(t))B_(n,k)(Dg(t),D^2g(t),...)
(3)

(M. Alekseyev, pers. comm., Nov. 3, 2006).

Faà di Bruno's formula can be cast in a framework that is a special case of a Hopf algebra (Figueroa and Gracia-Bondía 2005).

The first few derivatives for symbolic f and g are given by

d/(dt)f(g(t))=f^'(g(t))g^'(t)
(4)
(d^2)/(dt^2)f(g(t))=[g^'(t)]^2f^('')(g(t))+f^'(g(t))g^('')(t)
(5)
(d^3)/(dt^3)f(g(t))=3g^'(t)f^('')(g(t))g^('')(t)+[g^'(t)]^3f^((3))(g(t))+f^'(g(t))g^((3))(t).
(6)

See also

Derivative, Hopf Algebra, Leibniz Identity, Umbral Calculus

Explore with Wolfram|Alpha

References

Bertrand, J. Cours de calcul différentiel et intégral, tome I. Paris: Gauthier-Villars, p. 138, 1864.Cesàro, E. "Dérivées des fonctions de fonctions." Nouvelles Ann. 4, 41-45, 1885. Reprinted in Opere Scelte, Vol. 1. Rome: Edizioni Cremorese, pp. 416-429, 1964.Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 137-139, 1974.Dederick. "Successive Derivatives of a Function of Several Functions." Ann. Math. 27, 385-394, 1926.Faà di Bruno, C. F.. "Sullo sviluppo delle funzione." Ann. di Scienze Matem. et Fisiche di Tortoloni 6, 479-480, 1855.Faà di Bruno, C. F.. "Note sur un nouvelle formule de calcul différentiel." Quart. J. Math. 1, 359-360, 1857.Figueroa, H. and Gracia-Bondía, J. M. "Combinatorial Hopf Algebras in Quantum Field Theory I." 19 Mar 2005. http://arxiv.org/abs/hep-th/0408145.Français, J. F. "Analise transcendante. Du calcul des dérivations, ramené à ses véritables principes, ou théorie du développement des fonctions, et du retour des suites." Ann. Math. 6, 61-111, 1815.Johnson, W. P. "The Curious History of Faà di Bruno's Formula." Amer. Math. Monthly 109, 217-234, 2002.Joni, S. A. and Rota, C.-G. "The Faà di Bruno Bialgebra." §IX in "Coalgebras and Bialgebras in Combinatorics." Umbral Calculus and Hopf Algebras. Contemp. Math. 6, 18-21, 1982.Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, p. 33, 1965.Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 50, 1997.Marchand, E. "Sur le changement de variables." Ann. École Normale Sup. 3, 137-188 and 343-388, 1886.Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, pp. 35-37, 1958.Roman, S. "The Formula of Faa di Bruno." Amer. Math. Monthly 87, 805-809, 1980.Teixeira, F. G. "Sur les dérivées d'ordre quelconque." Giornale di Matem. di Battaglini 18, 306-316, 1880.Wall. "On the n-th Derivative of f(x)." Bull. Amer. Math. Soc. 44, 395-398, 1938.

Referenced on Wolfram|Alpha

Faà di Bruno's Formula

Cite this as:

Weisstein, Eric W. "Faà di Bruno's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FaadiBrunosFormula.html

Subject classifications