The combinatorics of higher derivatives of implicit functions

  • Shaul ZemelEmail author


We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.


Higher derivatives Implicit functions Combinatorial coefficients 

Mathematics Subject Classification

05A10 05A18 11B75 


  1. 1.
    Comtet, L.: Polynômes de Bell et Formule Explicite des Dérivées Successives d’une Fonction Implicite. C. R. Acad. Sci. Paris Sér. A tome 267, 457–460 (1968)zbMATHGoogle Scholar
  2. 2.
    Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions, p. xi+343. D. Reidel Publishing Company/Springer, Netherlands (1974)CrossRefGoogle Scholar
  3. 3.
    Comtet, L., Fiolet, M.: Sue les Dérivées Successives d’une Fonction Implicite. C. R. Acad. Sci. Paris Sér. A tome 278, 249–251 (1974)zbMATHGoogle Scholar
  4. 4.
    Figueroa, H., Garcia–Bondía, J. M., Várilly, J. C.: Faà di Bruno Hopf Algebras (pre-print).
  5. 5.
    Johnson, W.P.: The Curious History of Faà di Bruno’s Formula. Am. Math. Mon. 109(3), 217–234 (2002)zbMATHGoogle Scholar
  6. 6.
    Johnson, W.P.: Combinatorics of higher derivatives of inverses. Am. Math. Mon. 109(3), 273–277 (2002)CrossRefGoogle Scholar
  7. 7.
    Johnson, W.P.: Some problems in differentiation.
  8. 8.
    Nahay, J.M.: The \(n\)th order implicit differentiation formula for two variables with an application to computing all roots of a transcendental function. Math. Comput. Sci. 6(1), 79–105 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Schlank, T.M., Tessler, R.J., Zernik, A.: Exact maximum-entropy estimate for Feynman diagrams (pre-print). arXiv:1512.00752
  10. 10.
    Sokal, A.D.: A ridiculously simple and explicit implicit function theorem. Sémin. Lothar. Comb. 61A, 21 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Wilde, T.: Implicit higher derivatives, and a formula of Comtet and Fiolet (pre-print) (2008). arXiv:0805.2674
  12. 12.
    Worontzoff, M.: Sur le Développement en Séries des Fonctions Implicites. Nouv. Ann. Math. Sér. 3 tome 13, 167–184 (1894)zbMATHGoogle Scholar
  13. 13.
    Yuzhakov, A.P.: On an application of the multiple logarithmic residue to the expansion of implicit functions in power series. Mat. Sbornik 92(2), 177–192 (1975)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations