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 DerivativesExample 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 FunctionsMultivariable Functions 