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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 83, Issue 2
  • March 1978, pp. 159-165

Formulae for high derivatives of composite functions

  • L. E. Fraenkel (a1)
  • DOI:
  • Published online: 24 October 2008

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 = ℝM and F = ℝN, where ℝM 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 ℝM + 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).

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(5)J. Dieudonné Foundations of modern analysis (New York, Academic Press, 1969).

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