Question

Given u( x , y ) and y ( x , z ), show that

Short answer

The derivative of has been proved.

Step-by-step solution

Given data

The given expressions are u( x , y ) and y( x , z ) .

Concept of Partial Differentiation

The process of finding the partial derivative of a function is called partial differentiation. In this process, the partial derivative of a function with respect to one variable is found by keeping the other variable constant.

The formula to find the Partial derivative is given by:

Calculation of the product of (∂u/∂y)x and  (∂y/∂x)z

Since, the gas equation is given as follows:

So, the value of P can be written as follows:

Similarly, for the value of V as follows:

Calculation to solve the value of  (∂u/∂x)z

Substitute the values of P and V in the gas equation:

This is the specific heats in thermodynamics.

Now, obtain: