Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives.
It is named after Francesco Faà di Bruno (1855, 1857), although 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 had stated the formula in a calculus textbook,[1] which is considered to be the first published reference on the subject.
[2] Perhaps the most well-known form of Faà di Bruno's formula says that
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
leads to a somewhat simpler formula expressed in terms of partial (or incomplete) exponential Bell polynomials
: The formula has a "combinatorial" form: where The following is a concrete explanation of the combinatorial form for the
corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way.
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.
in the third line corresponds to the partition 2 + 2 of the integer 4, (4, because we are finding the fourth derivative), while
corresponds to the fact that there are two summands (2 + 2) in that partition.
The coefficient 3 corresponds to the fact that there are
ways of partitioning 4 objects into groups of 2.
variables are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables (if it seems opaque, see the very concrete example below):[3] where (as above)
More general versions hold for cases where the all functions are vector- and even Banach-space-valued.
The five terms in the following expression correspond in the obvious way to the five partitions of the set
is the number of parts in the partition: If the three variables are indistinguishable from each other, then three of the five terms above are also indistinguishable from each other, and then we have the classic one-variable formula.
or as an equivalent sum over integer partitions of
denoting the number of parts, or where is the set of partitions of
The first form is obtained by picking out the coefficient of
"by inspection", and the second form is then obtained by collecting like terms, or alternatively, by applying the multinomial theorem.
gives an expression for the reciprocal of the formal power series
Stanley[4] gives a version for exponential power series.
th derivative at 0: This should not be construed as the value of a function, since these series are purely formal; there is no such thing as convergence or divergence in this context.
This version of the formula is particularly well suited to the purposes of combinatorics.
are the same and are a factor common to every term: where
is the nth complete exponential Bell polynomial.
is a moment-generating function, and the polynomial in various derivatives of
is the polynomial that expresses the moments as functions of the cumulants.