Mixed partial derivative ( $f_{x y}$ )

A mixed partial derivative takes partial derivatives in different variables, for example

f_{x y} = \partial_{y} (\partial_{x} f)

. If both second partial derivatives are continuous in a neighbourhood, their order can be interchanged:

f_{x y} = f_{y x}

. The chain rule transforms mixed partial derivatives under a linear change of variables by composing the corresponding first-order differential operators.

Table of contents
- Unequal mixed partial derivatives Mixed partial derivative
- Symmetry of second derivatives Mixed partial derivative

Unequal mixed partial derivatives ( $\partial_{y} \partial_{x} f (0) = - 1, \partial_{x} \partial_{y} f (0) = 1$ )

For

f (x, y) = x y (x^{2} - y^{2}) / (x^{2} + y^{2})

away from the origin and zero at the origin,

∣ f ∣ \leq (x^{2} + y^{2}) /2

proves Fréchet differentiability with zero derivative. Yet

f_{x} (0, y) = - y

and

f_{y} (x, 0) = x

, giving the displayed unequal mixed partial derivatives. They exist but are not continuous at the origin, so Clairaut's theorem does not apply.

Symmetry of second derivatives ( $\partial_{i} \partial_{j} f = \partial_{j} \partial_{i} f$ )

When the second partial derivatives are continuous in a neighborhood, the order of the mixed partial derivatives can be interchanged. Expressing a rectangular finite difference as either iterated integral by the fundamental theorem of calculus and shrinking the rectangle proves the equality. Mere existence of both derivatives does not imply this symmetry.

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