This is a very strange question, and I'd like to be sure that I understand the notation before I try to answer it.

I'm guessing that those little subscripts indicate a quantity that is to be held constant. If so, then

, for example, makes good sense: z is a function of x and y, and when you take the partial derivative of z with respect to x you would naturally interpret this to mean that y is being held constant. But the next derivative,

, is another matter altogether. The variables r and ϑ go together, and when you form a partial derivative with respect to ϑ you would normally expect that r, not x, is the variable being held constant. To keep x constant when differentiating with respect to ϑ is unusual, not to say perverse. It can be done, but it needs to be done with care. Am I right in thinking that this is what this problem is really about, or am I misinterpreting it?