Given a function of two variables, ƒ ( *x, y*), the derivative with respect to *x* only (treating *y* as a constant) is called the **partial derivative of ƒ with respect to x **and is denoted by either ∂ƒ / ∂

*x*or ƒ

_{x}. Similarly, the derivative of ƒ with respect to

*y*only (treating

*x*as a constant) is called the

**partial derivative of ƒ with respect to**and is denoted by either ∂ƒ / ∂

*y**y*or ƒ

_{y}.

The second partial dervatives of f come in four types:

**Notations**

- Differentiate ƒ with respect to
*x*twice. (That is, differentiate ƒ with respect to*x*; then differentiate the result with respect to*x*again.)

- Differentiate ƒ with respect to
*y*twice. (That is, differentiate ƒ with respect to*y*; then differentiate the result with respect to*y*again.)

**Mixed partials:**

- First differentiate ƒ with respect to
*x*; then differentiate the result with respect to*y*.

- First differentiate ƒ with respect to
*y*; then differentiate the result with respect to*x*.

For virtually all functions ƒ ( *x*, *y*) commonly encountered in practice, ƒ _{vx }; that is, the order in which the derivatives are taken in the mixed partials is immaterial.

**Example 1:** If ƒ ( *x*, *y*) = 3 *x* ^{2} *y* + 5 *x* − 2 *y* ^{2} + 1, find ƒ _{x }, ƒ _{y }, ƒ _{xx }, ƒ _{yy }, ƒ _{xy }1, and ƒ _{yx }.

First, differentiating ƒ with respect to *x* (while treating *y* as a constant) yields

Next, differentiating ƒ with respect to *y* (while treating *x* as a constant) yields

The second partial derivative ƒ _{xx }means the partial derivative of ƒ _{x }with respect to *x*; therefore,

The second partial derivative ƒ _{yy }means the partial derivative of ƒ _{y }with respect to *y*; therefore,

The mixed partial ƒ _{xy }means the partial derivative of ƒ _{x }with respect to *y*; therefore,

The mixed partial ƒ _{yx }means the partial derivative of ƒ _{y }with respect to *x*; therefore,

Note that ƒ _{yx }= ƒ _{xy }, as expected.