Generalization of the formula of Faa di Bruno for a composite function with a vector argument

Abstract

The paper presents a new explicit formula for the nth total derivative of a composite function with a vector argument. The well-known formula of Faa di Bruno gives an expression for the nth derivative of a composite function with a scalar argument. The formula proposed represents a straightforward generalization of Faa di Bruno's formula and gives an explicit expression for the nth total derivative of a composite function when the argument is a vector with an arbitrary number of components. In this sense, the formula of Faa di Bruno is its special case. The mathematical tools used include differential operators, polynomials, and Diophantine equations. An example is shown for illustration.