Pure and Applied Mathematics Journal

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Partial Derivatives of Some Types of Two-Variables Functions

Received: 22 March 2013    Accepted:     Published: 02 April 2013
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Abstract

This paper mainly studies the evaluation of partial derivatives of four types of two-variables functions. We can obtain the infinite series forms of any order partial derivatives of these four types of functions by using differentiation term by term theorem, and hence reducing the difficulty of calculating their higher order partial derivative values greatly. On the other hand, we propose four functions of two-variables to evaluate their any order partial derivatives, and some of their higher order partial derivative values practically. At the same time, we employ Maple to calculate the approximations of these higher order partial derivative values and their infinite series forms for verifying our answers.

DOI 10.11648/j.pamj.20130202.12
Published in Pure and Applied Mathematics Journal (Volume 2, Issue 2, April 2013)
Page(s) 56-61
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Partial Derivatives, Differentiation Term By Term Theorem, Infinite Series Forms, Maple

References
[1] C. H., Bischof, G. Corliss, and A. Griewank, "Structured second and higher-order derivatives through univariate Taylor series," Optimization Methods and Software, 2, pp. 211-232, 1993.
[2] M. Hardy, "Combinatorics of partial derivatives," The Elec-tronic Journal of Combinatorics 13, #R1, 2006.
[3] L. E. Fraenkel, "Formulae for high derivatives of composite functions," Mathematical Proceedings of the Cambridge Philosophical Society, 83 : pp. 159-165, 1978.
[4] T-W, Ma, "Higher chain formula proved by combinatorics," The Electronic Journal of Combinatorics 16, #N21, 2009.
[5] D. N. Richard, "An efficient method for the numerical eval-uation of partial derivatives of arbitrary order," ACM Transactions on Mathematical Software (TOMS), 18(2), pp. 159-173, 1992.
[6] A. Griewank and A. Walther, Evaluating Derivatives:Principles and Techniques of Algorithmic Differentiation, 2nd ed., SIAM, 2008.
[7] P. Franklin, Methods of Advanced Calculus, McGraw-Hill Co., Inc., chap. II, 1944.
[8] L. Flatto, Advanced Calculus, The Williams & Wilkins Co., chap. 9, 1976.
[9] D. V. Widder, Advanced Calculus, 2 nd ed., Prentice-Hall, Inc, chap. 1&4., 1961.
[10] T. M. Apostol, Mathematical Analysis, 2nd ed., Addi-son-Wesley Publishing Co., Inc., p230, 1975.
Author Information
  • Department of Management and Information, Nan Jeon Institute of Technology, Tainan City, Taiwan

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  • APA Style

    Chii-Huei Yu. (2013). Partial Derivatives of Some Types of Two-Variables Functions. Pure and Applied Mathematics Journal, 2(2), 56-61. https://doi.org/10.11648/j.pamj.20130202.12

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    ACS Style

    Chii-Huei Yu. Partial Derivatives of Some Types of Two-Variables Functions. Pure Appl. Math. J. 2013, 2(2), 56-61. doi: 10.11648/j.pamj.20130202.12

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    AMA Style

    Chii-Huei Yu. Partial Derivatives of Some Types of Two-Variables Functions. Pure Appl Math J. 2013;2(2):56-61. doi: 10.11648/j.pamj.20130202.12

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  • @article{10.11648/j.pamj.20130202.12,
      author = {Chii-Huei Yu},
      title = {Partial Derivatives of Some Types of Two-Variables Functions},
      journal = {Pure and Applied Mathematics Journal},
      volume = {2},
      number = {2},
      pages = {56-61},
      doi = {10.11648/j.pamj.20130202.12},
      url = {https://doi.org/10.11648/j.pamj.20130202.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20130202.12},
      abstract = {This paper mainly studies the evaluation of partial derivatives of four types of two-variables functions. We can obtain the infinite series forms of any order partial derivatives of these four types of functions by using differentiation term by term theorem, and hence reducing the difficulty of calculating their higher order partial derivative values greatly. On the other hand, we propose four functions of two-variables to evaluate their any order partial derivatives, and some of their higher order partial derivative values practically. At the same time, we employ Maple to calculate the approximations of these higher order partial derivative values and their infinite series forms for verifying our answers.},
     year = {2013}
    }
    

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    AB  - This paper mainly studies the evaluation of partial derivatives of four types of two-variables functions. We can obtain the infinite series forms of any order partial derivatives of these four types of functions by using differentiation term by term theorem, and hence reducing the difficulty of calculating their higher order partial derivative values greatly. On the other hand, we propose four functions of two-variables to evaluate their any order partial derivatives, and some of their higher order partial derivative values practically. At the same time, we employ Maple to calculate the approximations of these higher order partial derivative values and their infinite series forms for verifying our answers.
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