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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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(1)Abraham, R. and Robbin, J.Transversal mappings and flows (New York, Benjamin, 1967).
(2)Bruno, F. de.Note sur une nouvelle formule de calcul différentiel. Quart. J. Math. 1 (1856), 359360.
(3)Cartan, H.Differential calculus (Paris, Hermann, and Boston, Houghton Mifflin, 1971).
(4)Chaundy, T.The differential calculus (Oxford, Clarendon Press, 1935).
(5)Dieudonné, J.Foundations of modern analysis (New York, Academic Press, 1969).
(6)Dieudonné, J.Treatise on analysis, vol. IV (New York, Academic Press, 1974).
(7)Goursat, E.Mathematical analysis, vol. I (Boston, Ginn, 1904).
(8)Stein, E. M.Singular integrals and differentiability properties of functions (Princeton, University Press, 1970).
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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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