# Second Order Partial Derivatives in Calculus

Examples with detailed solutions on how to calculate second order partial derivatives are presented.

## Definitions and Notations of Second Order Partial Derivatives

For a two variable function f(x , y), we can define 4 second order partial derivatives along with their notations.

## Examples with Detailed Solutions on Second Order Partial Derivatives

Example 1

Find f_{xx}, f_{yy} given that f(x , y) = sin (x y)

Solution

f_{xx} may be calculated as follows

f_{xx} = ?^{2}f / ?x^{2} = ?(?f / ?x) / ?x

= ?(?[ sin (x y) ]/ ?x) / ?x

= ?(y cos (x y) ) / ?x

= - y^{2} sin (x y) )

f_{yy} can be calculated as follows

f_{yy} = ?^{2}f / ?y^{2} = ?(?f / ?y) / ?y

= ?(?[ sin (x y) ]/ ?y) / ?y

= ?(x cos (x y) ) / ?y

= - x^{2} sin (x y) )
Example 2

Find f_{xx}, f_{yy}, f_{xy}, f_{yx} given that f(x , y) = x^{3} + 2 x y.

Solution

f_{xx} is calculated as follows

f_{xx} = ?^{2}f / ?x^{2} = ?(?f / ?x) / ?x

= ?(?[ x^{3} + 2 x y ]/ ?x) / ?x

= ?( 3 x^{2} + 2 y ) / ?x

= 6x

f_{yy} is calculated as follows

f_{yy} = ?^{2}f / ?y^{2} = ?(?f / ?y) / ?y

= ?(?[ x^{3} + 2 x y ]/ ?y) / ?y

= ?( 2x ) / ?y

= 0

f_{xy} is calculated as follows

f_{xy} = ?^{2}f / ?y?x = ?(?f / ?x) / ?y

= ?(?[ x^{3} + 2 x y ]/ ?x) / ?y

= ?( 3 x^{2} + 2 y ) / ?y

= 2

f_{yx} is calculated as follows

f_{yx} = ?^{2}f / ?x?y = ?(?f / ?y) / ?x

= ?(?[ x^{3} + 2 x y ]/ ?y) / ?x

= ?( 2x ) / ?x

= 2

Example 3

Find f_{xx}, f_{yy}, f_{xy}, f_{yx} given that f(x , y) = x^{3}y^{4} + x^{2} y.

Solution

f_{xx} is calculated as follows

f_{xx} = ?^{2}f / ?x^{2} = ?(?f / ?x) / ?x

= ?(?[ x^{3}y^{4} + x^{2} y ]/ ?x) / ?x

= ?( 3 x^{2}y^{4} + 2 x y) / ?x

= 6x y^{4} + 2y

f_{yy} is calculated as follows

f_{yy} = ?^{2}f / ?y^{2} = ?(?f / ?y) / ?y

= ?(?[ x^{3}y^{4} + x^{2} y ]/ ?y) / ?y

= ?( 4 x^{3}y^{3} + x^{2} ) / ?y

= 12 x^{3}y^{2}

f_{xy} is calculated as follows

f_{xy} = ?^{2}f / ?y?x = ?(?f / ?x) / ?y

= ?(?[ x^{3}y^{4} + x^{2} y ]/ ?x) / ?y

= ?( 3 x^{2}y^{4} + 2 x y ) / ?y

= 12 x^{2}y^{3} + 2 x

f_{yx} is calculated as follows

f_{yx} = ?^{2}f / ?x?y = ?(?f / ?y) / ?x

= ?(?[ x^{3}y^{4} + x^{2} y ]/ ?y) / ?x

= ?(4 x^{3}y^{3} + x^{2}) / ?x

= 12 x^{2}y^{3} + 2x

## More References and Links to Partial Derivatives and Multivariable Functions

Multivariable Functions