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Formulae for high derivatives of composite functions

  • L. E. Fraenkel (a1)

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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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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