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Partial Derivative


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.

 (partialf)/(partialx_m)=lim_(h->0)(f(x_1,...,x_m+h,...,x_n)-f(x_1,...,x_m,...,x_n))/h.
(1)

The above partial derivative is sometimes denoted f_(x_m) for brevity.

Partial derivatives can also be taken with respect to multiple variables, as denoted for examples

(partial^2f)/(partialx^2)=f_(xx)
(2)
(partial^2f)/(partialxpartialy)=f_(xy)
(3)
(partial^3f)/(partialx^2partialy)=f_(xxy).
(4)

Such partial derivatives involving more than one variable are called mixed partial derivatives.

For a "nice" two-dimensional function f(x,y) (i.e., one for which f, f_x, f_y, f_(xy), f_(yx) exist and are continuous in a neighborhood (a,b)), then

 f_(xy)(a,b)=f_(yx)(a,b).
(5)

More generally, for "nice" functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performed, so it also is true that

 f_(xxy)=f_(xyx)=f_(yxx).
(6)
PartialDerivative

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

 f(x,y)={(xy(x^2-y^2))/(x^2+y^2)   for (x,y)!=(0,0); 0   for (x,y)=(0,0),
(7)

which has f_(xy)(0,0)=-1 and f_(yx)(0,0)=1 (Wagon 1991). This function is depicted above and by Fischer (1986).

Abramowitz and Stegun (1972) give finite difference versions for partial derivatives.

A differential equation expressing one or more quantities in terms of partial derivatives is called a partial differential equation. Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve.


See also

Ablowitz-Ramani-Segur Conjecture, Derivative, Exact Differential, Mixed Partial Derivative, Monkey Saddle, Multivariable Calculus, Partial Differential Equation Explore this topic in the MathWorld classroom

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References

Abramowitz, M. and Stegun, I. 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 aus den Sammlungen von Universitäten und Museen, Bildband. 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.

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Partial Derivative

Cite this as:

Weisstein, Eric W. "Partial Derivative." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PartialDerivative.html

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