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Formulae for high derivatives of composite functions

Published online by Cambridge University Press:  24 October 2008

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
Mathematics Division, University of Sussex
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Extract

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).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 2 , March 1978 , pp. 159 - 165
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Abraham, R. and Robbin, J.Transversal mappings and flows (New York, Benjamin, 1967).Google Scholar
(2)Bruno, F. de.Note sur une nouvelle formule de calcul différentiel. Quart. J. Math. 1 (1856), 359–360.Google Scholar
(3)Cartan, H.Differential calculus (Paris, Hermann, and Boston, Houghton Mifflin, 1971).Google Scholar
(4)Chaundy, T.The differential calculus (Oxford, Clarendon Press, 1935).Google Scholar
(5)Dieudonné, J.Foundations of modern analysis (New York, Academic Press, 1969).Google Scholar
(6)Dieudonné, J.Treatise on analysis, vol. IV (New York, Academic Press, 1974).Google Scholar
(7)Goursat, E.Mathematical analysis, vol. I (Boston, Ginn, 1904).Google Scholar
(8)Stein, E. M.Singular integrals and differentiability properties of functions (Princeton, University Press, 1970).Google Scholar