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 x² y + 5 x − 2 y² + 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.

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