Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Formulae for high derivatives of composite functions

L. E. Fraenkela1

a1 Mathematics Division, University of Sussex

This note concerns a question of elementary calculus. Given a smooth composite function u = g o f [with values u(x) = g(f(x))], we write explicit formulae for its derivatives, of arbitrary order, in terms of derivatives of f and g. We consider (A) the general case,


in which E, F and G are Banach spaces, and U, V are open sets; (B) the finite-dimensional case E = xs211DM and F = xs211DN, where xs211DM denotes real M-dimensional Euclidean space; and (C) the particular case of (B) (due to restricting g to part of an M-dimensional surface in xs211DM + 1) in which N = M + 1 and u(x)= g(x, φ (x)), so that φ denotes a real-valued (scalar-valued) function of x = (x1, …, xM).

(Received September 12 1977)