The Derivatives of the Inverse of a One-to-One Multivariate Function

Abstract

Consider a multivariate function f that maps an open region of Rp into Rp, together with its inverse function. This paper presents a general method for obtaining the partial derivatives of the inverse function, of any order, expressed in terms of the partial derivatives of the inverse of the Jacobian matrix of f. The results are local. They require only that f have continuous partial derivatives up to the order of interest in a neighbourhood of a point, and a nonsingular Jacobian at that point, so global one-to-one-ness is not needed. A recursion for the derivatives of the inverse is proved by induction. Explicit formulas are derived for all partial derivatives up to the fourth order, both for the inverse function and for the inverse Jacobian. The symmetry of the mixed partial derivatives of the inverse is established. The computational cost is discussed: the explicit formulas grow factorially with the order, whereas a Leibniz-type recursion gives polynomial cost for numerical evaluation. A recursive scheme implemented in Python using symbolic computation is provided for derivatives of arbitrary order, together with a program that verifies the explicit formulas independently against direct differentiation. The univariate case, published previously by Withers, Nadarajah and Teal in the journal Axioms, is reviewed first, recovering known results expressed in terms of Bell polynomials. The method is illustrated by worked examples in two dimensions. Its use in mathematical statistics is shown by casting maximum likelihood estimation in canonical exponential families, including a canonical regression setting, in the framework of the paper, and by deriving explicitly the bias-corrected maximum likelihood estimator with bias On−2, which is checked against an exact calculation for the normal distribution.

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Cite

BibTeX RIS

BibTeX

@article{withers2026derivatives,
  title = {{The Derivatives of the Inverse of a One-to-One Multivariate Function}},
  author = {Christopher S. Withers and Saralees Nadarajah and Paul Teal},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100788},
  url = {https://www.mdpi.com/2075-1680/15/10/788},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - The Derivatives of the Inverse of a One-to-One Multivariate Function
AU  - Christopher S. Withers
AU  - Saralees Nadarajah
AU  - Paul Teal
PY  - 2026
DA  - 2026/10/21
JO  - Axioms
DO  - 10.3390/axioms15100788
UR  - https://www.mdpi.com/2075-1680/15/10/788
AB  - Consider a multivariate function f that maps an open region of Rp into Rp, together with its inverse function. This paper presents a general method for obtaining the partial derivatives of the inverse function, of any order, expressed in terms of the partial derivatives of the inverse of the Jacobian matrix of f. The results are local. They require only that f have continuous partial derivatives up to the order of interest in a neighbourhood of a point, and a nonsingular Jacobian at that point, so global one-to-one-ness is not needed. A recursion for the derivatives of the inverse is proved by induction. Explicit formulas are derived for all partial derivatives up to the fourth order, both for the inverse function and for the inverse Jacobian. The symmetry of the mixed partial derivatives of the inverse is established. The computational cost is discussed: the explicit formulas grow factorially with the order, whereas a Leibniz-type recursion gives polynomial cost for numerical evaluation. A recursive scheme implemented in Python using symbolic computation is provided for derivatives of arbitrary order, together with a program that verifies the explicit formulas independently against direct differentiation. The univariate case, published previously by Withers, Nadarajah and Teal in the journal Axioms, is reviewed first, recovering known results expressed in terms of Bell polynomials. The method is illustrated by worked examples in two dimensions. Its use in mathematical statistics is shown by casting maximum likelihood estimation in canonical exponential families, including a canonical regression setting, in the framework of the paper, and by deriving explicitly the bias-corrected maximum likelihood estimator with bias On−2, which is checked against an exact calculation for the normal distribution.
SN  - 2075-1680
ER  -

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