The general higher-order chain rule for differential calculus with applications

Mathematicians have investigated formulae for expressing higher-order derivatives of composite functions in terms of derivatives of their factor functions for over 200 years. This talk describes the formula for higher-order variational derivatives of composite functions, generalising the formula commonly attributed to Faa di Bruno to functions in locally convex topological vector spaces. The general structure of this result is required for dealing with composite functionals, which will be illustrated through applications in branching processes, quantum field theory, point process theory, Volterra series, and for multi-object tracking in signal processing applications.