Second Order Partial Derivatives in Calculus

Second Order Partial Derivatives in Calculus

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

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

2f / ∂x2 = ∂(∂f / ∂x) / ∂x or fxx

2f / ∂y2 = ∂(∂f / ∂y) / ∂y or fyy

2f / ∂x∂y = ∂(∂f / ∂y) / ∂x or fyx

2f / ∂y∂x = ∂(∂f / ∂x) / ∂y or fxy

Example 1: Find fxx, fyy given that f(x , y) = sin (x y)

solution to Example 1:

fxx may be calculated as follows

fxx = ∂2f / ∂x2 = ∂(∂f / ∂x) / ∂x

= ∂(∂[ sin (x y) ]/ ∂x) / ∂x

= ∂(y cos (x y) ) / ∂x

= - y2 sin (x y) )

fyy can be calculated as follows

fyy = ∂2f / ∂y2 = ∂(∂f / ∂y) / ∂y

= ∂(∂[ sin (x y) ]/ ∂y) / ∂y

= ∂(x cos (x y) ) / ∂y

= - x2 sin (x y) )

Example 2: Find fxx, fyy, fxy, fyx given that f(x , y) = x3 + 2 x y.

solution to Example 2:

fxx is calculated as follows

fxx = ∂2f / ∂x2 = ∂(∂f / ∂x) / ∂x

= ∂(∂[ x3 + 2 x y ]/ ∂x) / ∂x

= ∂( 3 x2 + 2 y ) / ∂x

= 6x

fyy is calculated as follows

fyy = ∂2f / ∂y2 = ∂(∂f / ∂y) / ∂y

= ∂(∂[ x3 + 2 x y ]/ ∂y) / ∂y

= ∂( 2x ) / ∂y

= 0

fxy is calculated as follows

fxy = ∂2f / ∂y∂x = ∂(∂f / ∂x) / ∂y

= ∂(∂[ x3 + 2 x y ]/ ∂x) / ∂y

= ∂( 3 x2 + 2 y ) / ∂y

= 2

fyx is calculated as follows

fyx = ∂2f / ∂x∂y = ∂(∂f / ∂y) / ∂x

= ∂(∂[ x3 + 2 x y ]/ ∂y) / ∂x

= ∂( 2x ) / ∂x

= 2

Example 3: Find fxx, fyy, fxy, fyx given that f(x , y) = x3y4 + x2 y.

solution to Example 2:

fxx is calculated as follows

fxx = ∂2f / ∂x2 = ∂(∂f / ∂x) / ∂x

= ∂(∂[ x3y4 + x2 y ]/ ∂x) / ∂x

= ∂( 3 x2y4 + 2 x y) / ∂x

= 6x y4 + 2y

fyy is calculated as follows

fyy = ∂2f / ∂y2 = ∂(∂f / ∂y) / ∂y

= ∂(∂[ x3y4 + x2 y ]/ ∂y) / ∂y

= ∂( 4 x3y3 + x2 ) / ∂y

= 12 x3y2

fxy is calculated as follows

fxy = ∂2f / ∂y∂x = ∂(∂f / ∂x) / ∂y

= ∂(∂[ x3y4 + x2 y ]/ ∂x) / ∂y

= ∂( 3 x2y4 + 2 x y ) / ∂y

= 12 x2y3 + 2 x

fyx is calculated as follows

fyx = ∂2f / ∂x∂y = ∂(∂f / ∂y) / ∂x

= ∂(∂[ x3y4 + x2 y ]/ ∂y) / ∂x

= ∂(4 x3y3 + x2) / ∂x

= 12 x2y3 + 2x



More on partial derivatives and mutlivariable functions.
Multivariable Functions

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