Why Does the Chain Rule in Thermodynamics Include a Negative Sign?

[korialstasz]

I'm trying to understand why

$$\left(\frac{\partial P}{\partial T}\right)_V = -\left(\frac{\partial P}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_P$$

where does the minus sign come from?

 

[Chestermiller]

What is dV(P,T) in terms of dP and dT?