Q. 16 - Calculus

Question

Assume that g(x, y, z) is a function of three variables with partial derivatives of every order. Assume also the order in which the partial derivatives are taken is significant. 

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

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

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

Step-by-Step Solution

Answer

Part (a) The number of the second-order partial derivative is 9.

Part (b)The number of third order partial derivative is 27.

Part (c) The number of n-th order partial derivatives is $3^n$.

Step 1: Given information

We are given a function g of three variable x ,y, z

Part (a) Step 1: Explanation

As g is a function of three variables x, y, z.

 

The second-order partial derivatives can be given as,

$f_{xx},f_{xy},f_{xz},f_{yx},f_{yz},f_{yy},f_{zx},f_{zy},f_{zz}$.

It has 9 second-order partial derivatives.

Part (b) Step 1: Explanation

As g is a function of three variables x, y, z.

It will have 27 third-order partial derivatives.

Part (c) Step 1: Explanation.

As g is a function of x, y, z The number of n-th partial derivatives is $3^n$.