I don't really understand this calc question What do you even do here? Take the other two variable to RHS?


Well, you have $$z^z = cx^{-x}y^{-y}$$, then $z\log z = \log c-x\log x - y\log y$.
So $$\frac{\partial z}{\partial x}\log z + \frac{\partial z}{\partial x} = -\log x - 1.$$
Then, $$\frac{\partial z}{\partial x} = -\frac{1+\log x}{1+\log z}$$
