Faà di Bruno's formula

Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named after Francesco Faà di Bruno (1855, 1857), though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast stated the formula in a calculus textbook, considered the first published reference on the subject.

Perhaps the most well-known form of Faà di Bruno's formula says that

where the sum is over all n-tuples of nonnegative integers (m₁, …, m_n) satisfying the constraint

Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:

Combining the terms with the same value of m₁ + m₂ + ... + m_n = k and noticing that m _j has to be zero for j > n − k + 1 leads to a somewhat simpler formula expressed in terms of Bell polynomials B_n,k(x₁,...,x_n−k+1):

The formula has a "combinatorial" form:

where

The combinatorial form may initially seem forbidding, so let us examine a concrete case, and see what the pattern is:

The pattern is

The factor ${\displaystyle \scriptstyle g''(x)g'(x)^{2}\;}$ corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way. The factor ${\displaystyle \scriptstyle f'''(g(x))\;}$ that goes with it corresponds to the fact that there are three summands in that partition. The coefficient 6 that goes with those factors corresponds to the fact that there are exactly six partitions of a set of four members that break it into one part of size 2 and two parts of size 1.

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