The Equality of Mixed Partial Derivatives

Suppose thatis a function ofandwith partial derivatives andThese are again functions ofandand may themselves possess partial derivatives:

and

The middle two terms are called the second order partials. There are two mixed partials - andThe first is obtained by differentiatingfirst with respect tothenand the second is obtained by differentiating first with respect tothen

For many functionsIn fact ifand it;s partialsand are all continuous on a setthenon

Ifis a function of three variablesthen there are three first partialsandand nine second partials:and

Again the second partials are equal:

and

provided thatand all it's first and second partial derivatives are continuous.

Example:

Hence