Verified answer

This is a general properties of partial derivatives of functions of more than one variable; it's not just restricted to state functions.

Take any three variables that are related by some well-behaved function f(x,y,z) = 0. That means we can always rewrite this function as z = g(x,y), or y = h(x,z), or z = k(y,z), where g, h, and k are other well-behaved functions.

Let x be a function of y and z. Take the total differential of x:

dx = (∂x/∂y)_z dy + (∂x/∂z)_y dz

Take the total differential of y:

dy = (∂y/∂x)_z dx + (∂y/∂z)_x dz

Now substute the expression for dy in this last equation for dy in the first:

dx = (∂x/∂y)_z *[(∂y/∂x)_z dx + (∂y/∂z)_x dz] + (∂x/∂z)_y dz

dx = (∂x/∂y)_z * (∂y/∂x)_z dx + (∂x/∂y)_z * (∂y/∂z)_x dz + (∂x/∂z)_y dz

0 = [(∂x/∂y)_z * (∂y/∂x)_z - 1] dx + [(∂x/∂y)_z * (∂y/∂z)_x + (∂x/∂z)_y] dz

Because x and z are independent variables, and can be changed at will, both of the factors in brackets must independently be equal to zero in order for this equation to be true. This implies that:

(∂x/∂y)_z * (∂y/∂x)_z = 1

and

(∂x/∂y)_z * (∂y/∂z)_x + (∂x/∂z)_y = 0

-(∂x/∂y)_z * (∂y/∂z)_x = (∂x/∂z)_y

-(∂x/∂y)_z /(∂x/∂z)_y = 1/(∂y/∂z)_x

Using the previous result, we can also write this as:

-(∂x/∂y)_z /(∂x/∂z)_y = (∂z/∂y)_x

This is a general result. In your case, let x = V, y = T, and z = p, then:

-(∂V/∂T)_p /(∂V/∂p)_T = (∂p/∂T)_V

Which is what you are supposed to show.

PV = nRT P, T, n, R are given, discover V differentiate with appreciate to time t, be conscious that n and R are fixed P(dV/dt) + V(dP/dt) = nR (dT/dt) P, dP/dt, dT/dt, n and R are all given, and you merely got here upon V, discover dV/dt.