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# Partial derivatives of higher order.

If the partial derivatives of f are differentiable, we get the second order derivatives of f, and so on. We have:

Example 5.13   Let . Then:

Notations: , , and .

Theorem 5.14   Let f(x,y) be a function of two variables having partial derivatives fx, fy, fxy and fyx on an open domain in . If and if the partial derivatives are all continuous at (x0,y0), then

fxy(x0,y0)=fyx(x0,y0)

As a first example, see Ex.  5.13.

Example 5.15   Let . Then:

The computation of the mixed second derivative, in reversed order, is much more complicated:

We can iterate the above description and define the partial derives of order higher than 2, namely , , , and so on.

Noah Dana-Picard
2001-05-30