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 y and is denoted by either ∂ƒ / ∂ 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.