Partial Derivative Proof (thermodynamics notation)

Show that:

I have Euler's chain rule and "the splitter." Also the property, called the "inverter" where you can reciprocate a partial derivative.

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:

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:

However, I do not know what to do with this because I have the term

, 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.)