Q. 15

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 significant.

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

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

Step-by-Step Solution

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Answer

Part (a) The number of 2 order partial derivative is 4.

Part (b) the number of 3rd order of partial derivative is 8.

Part (c) the number of n-th order partial derivative is $2^{n}$ .

1Step 1: Given information

We are given a function f which is a function of two variables x and y

2Part (a) Step 1: Explanation

As f is a function of two variable x and y

the second order partial derivatives can be given as

$f_{x x}, f_{x y}, f_{y x}, f_{y y}$ . Therefore it has 4 second order partial derivatives

3Part (b) Step 1: Explanation

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

The third-order partial derivatives can be given as,

f_{x x x}, f_{x x y}, f_{x y x}, f_{y x x}, f_{x y y}, f_{y x y}, f_{y y x}, f_{y y y}

It has 8 third-order partial derivatives.

4Part (c) Step 1: Explanation.

As f is a function of 2 variables x and y,

the number of n-th partial derivatives is $2^{n}$ .

Q. 13

Q. 16

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