# 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 DerivativesFor a two variable function f(x , y), we can define 4 second order partial derivatives along with their notations.
## Examples with Detailed Solutions## Example 1Find f_{xx}, f_{yy} given that f(x , y) = sin (x y)
solution to Example 1:f _{xx} may be calculated as followsf _{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 followsf _{yy} = ∂^{2}f / ∂y^{2} = ∂(∂f / ∂y) / ∂y
= ∂(∂[ sin (x y) ]/ ∂y) / ∂y = ∂(x cos (x y) ) / ∂y = - x ^{2} sin (x y) )
## Example 2Find f_{xx}, f_{yy}, f_{xy}, f_{yx} given that f(x , y) = x^{3} + 2 x y.
solution to Example 2:f _{xx} is calculated as followsf _{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 followsf _{yy} = ∂^{2}f / ∂y^{2} = ∂(∂f / ∂y) / ∂y
= ∂(∂[ x ^{3} + 2 x y ]/ ∂y) / ∂y
= ∂( 2x ) / ∂y = 0 f _{xy} is calculated as followsf _{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 followsf _{yx} = ∂^{2}f / ∂x∂y = ∂(∂f / ∂y) / ∂x
= ∂(∂[ x ^{3} + 2 x y ]/ ∂y) / ∂x
= ∂( 2x ) / ∂x = 2
## Example 3Find f_{xx}, f_{yy}, f_{xy}, f_{yx} given that f(x , y) = x^{3}y^{4} + x^{2} y.
solution to Example 2:f _{xx} is calculated as followsf _{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 followsf _{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 followsf _{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 followsf _{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 on partial derivatives and mutlivariable functions. Multivariable Functions |