# Partial Derivative Proof (thermodynamics notation)

 P: 239 1. The problem statement, all variables and given/known data Show that: $$\left(\frac{\partial z}{\partial y}\right)_{u} = \left(\frac{\partial z}{\partial x}\right)_{y} \left[ \left(\frac{\partial x}{\partial y}\right)_{u} - \left(\frac{\partial x}{\partial y}\right)_{z} \right]$$ 2. Relevant equations I have Euler's chain rule and "the splitter." Also the property, called the "inverter" where you can reciprocate a partial derivative. 3. The attempt at a solution If I write Euler's chain rule, I only know how to write it when there are 3 variables, I usually write it in the form: $$\left(\frac{\partial x}{\partial y}\right)_{z} \left(\frac{\partial y}{\partial z}\right)_{x} \left(\frac{\partial z}{\partial x}\right)_{y} = -1$$ Where I can write x,y,z in any order as long as each variable is used in every spot. However, I do not know how to work this chain rule if I have an extra variable (u in this case). I also tried using the "splitter" to do something like writing: $$\left(\frac{\partial z}{\partial y} \right)_{u} = \left(\frac{\partial z}{\partial x} \right)_{u} \left(\frac{\partial x}{\partial y}\right)_{u}$$ However, I do not know what to do with this because I have the term $$\left(\frac{\partial z}{\partial x} \right)_{u}$$ , which doesn't appear in the original problem. Any help would be appreciated. Thanks in advance. (This is for a thermodynamics course, but we are still in the mathematics introduction.)