2 Second partial derivatives

Performing two successive partial differentiations of f ( x , y ) with respect to x (holding y constant) is denoted by 2 f x 2 (or f x x ( x , y ) ) and is defined by

2 f x 2 x f x

For functions of two or more variables as well as 2 f x 2 other second-order partial derivatives can be obtained. Most obvious is the second derivative of f ( x , y ) with respect to y is denoted by 2 f y 2 (or f y y ( x , y ) ) which is defined as:

2 f y 2 y f y

Example 5

Find 2 f x 2 and 2 f y 2 for f ( x , y ) = x 3 + x 2 y 2 + 2 y 3 + 2 x + y .

Solution

f x = 3 x 2 + 2 x y 2 + 0 + 2 + 0 = 3 x 2 + 2 x y 2 + 2

2 f x 2 x f x = 6 x + 2 y 2 + 0 = 6 x + 2 y 2 .

f y = 0 + x 2 × 2 y + 6 y 2 + 0 + 1 = 2 x 2 y + 6 y 2 + 1

2 f y 2 = y f y = 2 x 2 + 12 y .

We can use the alternative notation when evaluating derivatives.

Example 6

Find f x x ( 1 , 1 ) and f y y ( 2 , 2 ) for f ( x , y ) = x 3 + x 2 y 2 + 2 y 3 + 2 x + y .

Solution

f x x ( 1 , 1 ) = 6 × ( 1 ) + 2 × ( 1 ) 2 = 4.

f y y ( 2 , 2 ) = 2 × ( 2 ) 2 + 12 × ( 2 ) = 16

2.1 Mixed second derivatives

It is possible to carry out a partial differentiation of f ( x , y ) with respect to x followed by a partial differentiation with respect to y (or vice-versa). The results are examples of mixed derivatives . We must be careful with the notation here.

We use 2 f x y to mean ‘differentiate first with respect to y and then with respect to x ’ and we use 2 f y x to mean ‘differentiate first with respect to x and then with respect to y ’:

i.e. 2 f x y x f y and 2 f y x y f x .

(This explains why the order is opposite of what we expect - the derivative ‘operates on the left’.)

Example 7

For f ( x , y ) = x 3 + 2 x 2 y 2 + y 3 find 2 f x y .

Solution

f y = 4 x 2 y + 3 y 2 ; 2 f x y = 8 x y

The remaining possibility is to differentiate first with respect to x and then with respect to y i.e. y f x .

For the function in Example 7 f x = 3 x 2 + 4 x y 2   and 2 f y x = 8 x y . Notice that for this function

2 f x y 2 f y x .

This equality of mixed derivatives is true for all functions which you are likely to meet in your studies.

To evaluate a mixed derivative we can use the alternative notation. To evaluate 2 f x y we write f y x ( x , y ) to indicate that the first differentiation is with respect to y . Similarly, 2 f y x is denoted by f x y ( x , y ) .

Example 8

Find f y x ( 1 , 2 ) for the function f ( x , y ) = x 3 + 2 x 2 y 2 + y 3

Solution

f x = 3 x 2 + 4 x y 2  and f y x = 8 x y so f y x ( 1 , 2 ) = 8 × 1 × 2 = 16.

Task!

Find f x x ( 1 , 2 ) , f y y ( 2 , 1 ) , f x y ( 3 , 3 ) for   f ( x , y ) x 3 + 3 x 2 y 2 + y 2 .

f x = 3 x 2 + 6 x y 2 ; f y = 6 x 2 y + 2 y

f x x = 6 x + 6 y 2 ; f y y = 6 x 2 + 2 ; f x y = f y x = 12 x y

f x x ( 1 , 2 ) = 6 + 24 = 30 ; f y y ( 2 , 1 ) = 26 ; f x y ( 3 , 3 ) = 108