Faà di Bruno's Formula

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


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


(Roman 1980).

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


(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


See also

Derivative, Hopf Algebra, Leibniz Identity, Umbral Calculus

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Faà di Bruno's Formula

Cite this as:

Weisstein, Eric W. "Faà di Bruno's Formula." From MathWorld--A Wolfram Web Resource.

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