A Review of Francesco Faà di Bruno’s Formula and Life: Engineering, Mathematics, Faith and Divulgation

Probably the most famous result obtained by the Italian mathematician, engineer, inventor, musician, architect and priest Francesco Faà di Bruno (1825-1888) is a formula for the m-th derivative of a composite, real-valued function G(t)=g(f(t)), generalizing the well-known chain rule, and mentioned in books of mathematical statistics, combinatorial analysis, matrix theory, finite differential calculus, computer science, partitions, variational calculus and stochastic processes. This paper revisits the history and developments of Faà di Bruno’s formula, showing a simple numerical example to demonstrate the usefulness of his generalized approach and providing an overview of the main applications, from mathematical and engineering fields up to the use of the formula in symbolic computational engines like Matlab® and Wolfram (formerly known as Mathematica). This work also depicts a historical portrait of the scientist, who embodied in the 19th century a unique synthesis of scientific ingenuity, social engagement and religious fervor.