Can anyone answer this?
Well, as you suggested, it's not very clear what f is supposed to be. My best guess would be that f is supposed to be z (because it was defined as a function of x and y) and that they're asking for implicit differentiation. You could also isolate z and then use a quotient rule, which would get you to the same place.
For example, if you take the partial in x you get
If you want to get rid of the z, you can write
and then substitute back in to get
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