Sum of partial derivatives for an implicit function?

Question

This seems like a very easy question, but I dont know the rules of partial differentiation of implicit functions. I want to understand not only the solution to this particular problem but also how I should approach such implicit partial differentiation problems.

$$u = \phi(x-y, y-z,z-x) $$

Then $$\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = ?$$

Answer 1

Hint. Set $p=x-y,\,q=y-z$ and $r=z-x.$ Then you find $u_x$ by $$u_pp_x+u_qq_x+u_rr_x.$$ Proceed similarly for the other derivatives, then sum them up.