Q. 17 - Calculus

Question

Assume that f(x, y) is a function of two variables with partial derivatives of every order. Assume also that the order in which the partial derivatives are taken is not significant. 

(a) How many different second-order partial derivatives does f have? 

(b) How many different third-order partial derivatives does f have? 

(c) How many different nth-order partial derivatives does f have? 

Step-by-Step Solution

Answer

Part (a) It has 3-second order partial derivatives.

Part (b) It has 4 third-order partial derivatives.

Part (c) It has n-1 n-th order partial derivatives.

Step 1: Given information

We are given a function f of two variable x and y.

Part (a) Step 1: Explanation

As f is a function of two variables the second-order partial derivatives can be given as,

$f_{xx},f_{xy},f_{yx},f_{yy}$ as the order in which the derivative is taken does not matter hence the number of the partial derivatives is 3.

Part b) Step 1: Explanation

As f is a function of two variables x and y.

The third-order partial derivative can be given as 

$f_{xxx},f_{xxy},f_{xyx},f_{yxx},f_{xyy},f_{yyx},f_{yxy},f_{yyy},$

As the order of the derivative does not matter.

Hence it has 4 partial derivatives of third-order.

Part (c) Step 1: Explanation

As f is a function of two variables x and y also the order in which the derivative is taken does not matter hence 

It has n-1 n-th partial derivatives