Partial Derivatives of Some Types of Two-Variables Functions

Chii-Huei Yu

doi:doi:10.11648/j.pamj.20130202.12

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Partial Derivatives of Some Types of Two-Variables Functions

Chii-Huei Yu

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 2)

Published: 2 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.

Published in	Pure and Applied Mathematics Journal (Volume 2, Issue 2)
DOI	10.11648/j.pamj.20130202.12
Page(s)	56-61
Creative Commons	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), 2013. 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.

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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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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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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://article.sciencepublishinggroup.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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TY  - JOUR
T1  - Partial Derivatives of Some Types of Two-Variables Functions
AU  - Chii-Huei Yu
Y1  - 2013/04/02
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130202.12
DO  - 10.11648/j.pamj.20130202.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 56
EP  - 61
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130202.12
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.
VL  - 2
IS  - 2
ER  -

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Author Information

Chii-Huei Yu

Department of Management and Information, Nan Jeon Institute of Technology, Tainan City, Taiwan

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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

Copy | Download

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

Copy | Download

@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://article.sciencepublishinggroup.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}
}

Copy | Download

TY  - JOUR
T1  - Partial Derivatives of Some Types of Two-Variables Functions
AU  - Chii-Huei Yu
Y1  - 2013/04/02
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130202.12
DO  - 10.11648/j.pamj.20130202.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 56
EP  - 61
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130202.12
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.
VL  - 2
IS  - 2
ER  -

Copy | Download