Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.

(1) |

For nice functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performed
so, for example,

(2) |

(3) |

(4) |

(5) |

If the continuity requirement for Mixed Partials is dropped, it is possible to construct functions for which
Mixed Partials are *not* equal. An example is the function

(6) |

Abramowitz and Stegun (1972) give Finite Difference versions for partial derivatives.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 883-885, 1972.

Fischer, G. (Ed.). Plate 121 in
*Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.*
Braunschweig, Germany: Vieweg, p. 118, 1986.

Thomas, G. B. and Finney, R. L. §16.8 in *Calculus and Analytic Geometry, 9th ed.* Reading, MA: Addison-Wesley, 1996.

Wagon, S. *Mathematica in Action.* New York: W. H. Freeman, pp. 83-85, 1991.

© 1996-9

1999-05-26