Mathematics in Computer Science

, Volume 6, Issue 1, pp 79–105 | Cite as

The nth Order Implicit Differentiation Formula for Two Variables with an Application to Computing All Roots of a Transcendental Function



We prove a version of a generalization of the Lagrange inversion formula (LIF) for an implicit equation G(z, w) = 0 of two variables, expressing the nth derivative of z with respect to w as a polynomial in the mixed partial derivatives function of G with respect to z and w, and negative powers of the separant \({G_z \equiv \frac{\partial G}{\partial z}}\). Our method of proof is original, using only induction, and hence requires only that G be n-times differentiable in both variables, and requires only that the separant be nonzero. We then move on to a novel application of this LIF-like formula to derive a power series formula for each of the countably infinitely many roots of a pseudopolynomial—a finite sum of powers of a variable but allowing the powers to be any complex numbers.


Lagrange inversion Implicit differentiation Transcendental 

Mathematics Subject Classification (2010)

70H03 34K32 34A09 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bartholomew-Biggs M.C.: Using forward accumulation for automatic differentiation of implicitly-defined functions. Comput. Optim. Appl. 9, 65–84 (1998)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bender E.A., Bruce Richmond L.: A multivariate Lagrange inversion formula for asymptotic calculations. Electron. J. Comb 5, 1–4 (1998)Google Scholar
  3. 3.
    Bernardini A., Natalini P., Ricci P.E.: Multidimensional bell polynomials of higher order. Comput. Math. Appl. 0, 1–11 (2005)MathSciNetGoogle Scholar
  4. 4.
    Cayley, A.: Note on a differential equation. In: Manchester Society Proceedings, pp. 111–114 (1862)Google Scholar
  5. 5.
    Chou W.S., Hsu L.C., Shiue P.J.-S.: Application of Faa di Bruno’s formula in characterization of inverse relations. J. Comput. Appl. Math. 190, 151–169 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Christianson B.: Reverse accumulation and implicit functions. Optim. Methods Softw. 9, 307–322 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Constantine G.M., Savits T.H.: A multivariable Faa di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Craik A.D.D.: Prehistory of Faa di Bruno’s formula. Am. Math. Monthly. 112(2), 119–130 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Egorychev, G.P.: Integral Representation and the Computation of Combinatorial Sums. In: Translations of Mathematical Monographs, American Mathematical Society, vol. 59 (1984)Google Scholar
  10. 10.
    Faa di Bruno F.: Sullo sviluppo delle Funzioni. Annal di Scienze Mathematiche e Fisiche. 6, 479–480 (1855)Google Scholar
  11. 11.
    Faa di Bruno F.: Note sur une nouvelle formule de calcul differentiel. Q. J. Math. 1, 359–360 (1857)Google Scholar
  12. 12.
    Gallop, E.G.: Change of the independent variable in a differential coefficient. Trans. Camb. Phil. Soc. 16, 116–132Google Scholar
  13. 13.
    Hardy G.H., Ramanujan S.: Asymptotic formula in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–115 (1918)CrossRefGoogle Scholar
  14. 14.
    Johnson W.P.: The curious history of Faa di Bruno’s formula. Am. Math. Monthly. 109(3), 217–234 (2002)MATHCrossRefGoogle Scholar
  15. 15.
    Kamber F.: Formules exprimant les valeurs des coefficients des series de puissances inverses. Acta Mathematica 78, 193–204 (1946)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lagrange J.-L.: Nouvelle methode pour resoudre les equations litterales par le moyen des series. Memoires de l’Academie Royale des Sciences et Belles-Lettres de Berlin. 24, 251–326 (1770)Google Scholar
  17. 17.
    Ma T.-Y.: Higher chain formula proved by combinatorics. Electron. J. Comb 21, 1–7 (2009)Google Scholar
  18. 18.
    Mishkov R. L.: Generalization of the formula of Faa Di Bruno for a composite function with a vector argument. Int. J. Math. Math. Sci. 24(7), 481–491 (2000)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Mumford D.: Tata Lectures on Theta II. Birkhauser, Basel (1993)Google Scholar
  20. 20.
    Murphy R.: First memoir on the theory of analytical operations. Phil. Trans. Royal Soc. Lond. 127, 179–210 (1837)CrossRefGoogle Scholar
  21. 21.
    Nahay, J.M.: Linear differential resolvents. Dissertation. Rutgers University, Piscataway (2000)Google Scholar
  22. 22.
    Scherk, H.F.: De evolvenda functione \({\frac{yd.yd.yd \ldots ydx}{dx^{n}}}\) disquisitions nonnullae analyticae. Dissertation. Friedrich-Wilhelms-Universitat, Berlin (1823)Google Scholar
  23. 23.
    Sylvester J.J.: Note on Burman’s law for the inversion of the independent variable. Philosophical Magazine 8, 535–540 (1854) (4th series)Google Scholar
  24. 24.
    Wagner M., Schaefer B.-J., Walther A.: On the efficient computation of high-order derivatives for implicitly defined functions. Comput. Phys. Commun. 181, 756–764 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Broadway Performance SystemsColumbusUSA

Personalised recommendations