Faà di Bruno's formula

Faà di Bruno's formula expresses the nth derivative of a composite function as a sum involving derivatives of the outer and inner functions. Partial Bell polynomials organize its terms.

Partial Bell polynomials: Polynomials that encode partitions of a set into blocks of specified sizes. They collect the products of inner-function derivatives in each term.
First derivative: The derivative describing a function's instantaneous rate of change. The order-one case reduces directly to the ordinary chain rule.
Francesco Faà di Bruno: An Italian mathematician and priest whose work includes a formula for derivatives of composite functions. His name became attached to the formula for higher derivatives of compositions.
Moments and cumulants: Numerical summaries of a probability distribution and related quantities that encode its shape and dependence structure. Derivatives of moment-generating functions connect the formula to relations among these quantities.
Faà di Bruno's formula for inverse functions: A specialization of Faà di Bruno's formula expressing derivatives of an inverse function through derivatives of the original. It applies the general composition rule to the identity formed by a function and its inverse.
Bell polynomials: Polynomial sequences that encode set partitions and arise in combinatorics and probability. The partial Bell polynomials used here belong to this broader family.
Second derivative: The derivative of a function's first derivative, measuring how its rate of change varies. Its two terms first show distinct ways derivatives can fall on a composition.
Louis François Antoine Arbogast: A French mathematician who developed methods for calculating derivatives of composite functions. His earlier work on higher derivatives anticipated the formula later associated with Faà di Bruno.
Moment-generating function: A function whose derivatives at zero, when they exist, give the moments of a probability distribution. Differentiating its exponential representation invokes the formula's partition structure.
Lagrange inversion theorem: A theorem for extracting coefficients of compositional inverses of formal power series. It addresses inverse composition through coefficients rather than direct derivative expansion.

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