Partial Differentiation

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 _xy1, 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 _xxmeans the partial derivative of ƒ _xwith respect to x; therefore,

The second partial derivative ƒ _yymeans the partial derivative of ƒ _ywith respect to y; therefore,

The mixed partial ƒ _xymeans the partial derivative of ƒ _xwith respect to y; therefore,

The mixed partial ƒ _yxmeans the partial derivative of ƒ _ywith respect to x; therefore,

Note that ƒ _yx= ƒ _xy, as expected.

Differential Equations

Partial Differentiation