Dear Rajorshi Koyal,

If z=f(x,y) is a smooth function where

z is the dependent variable, x and y are the independent variables.

we have

dz=(∂f/∂x)dx +(∂f/∂y)dy, it is called the total differentiation.

where(∂f/∂x) is the partial derivative of f(x,y) with respect to x

(∂f/∂y) is the partial derivative of f(x,y) with respect to y.

In your question,

We have V = V(P, T), V depends on the variables P and T

and then

dV=(∂V/∂P)Tdp+(∂V/∂T)PdT ....(1`)

Assuming that P varies between P₁ and P₂ and T varies between T₁ and T₂, Integrating both sides with your initial conditions, the required forms yields.

V=delta= ∫(∂V/∂P)Tdp+∫(∂V/∂T)PdT..... (2)

the boundaries of p from P₁ to P₂

he boundaries of T from T₁ to T₂

where (∂V/∂P)T computed at the value T (T₁ or T₂ )

and (∂V/∂T)P computed at the value P (P₁ or P₂ )

Integrating (1) we obtain (2) and differentiating (2) we obtain(1).

The choice based on your initial conditions of the mathematical model.

Best regards

Just to be sure, could we have a copy of the previous part as well where equations (1.1) and (1.2) are given?

Maybe the attachement helps.