Skip to main content
×
Home

Formulae for high derivatives of composite functions

  • L. E. Fraenkel (a1)
Abstract

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).

Copyright
References
Hide All
(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).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 53 *
Loading metrics...

Abstract views

Total abstract views: 653 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd November 2017. This data will be updated every 24 hours.