Your last expression is close to correct. I suspect that notation is getting in the way as well. We already have a function f(x, y, x) so using it in the quotient rule is a little confusing. So I am going to say

And I am going to define the quotient rule as

given a function

then

So to get the derivative then choose m(x, y, z) = x^3+3xy^2-3xyz-z^4 and n(x) = x^2 + z^2

See what you can do with that.

-Dan