Will this stump you?

[MangoTBG]

If f is a differentiable function of ONE VARIABLE, and z(x,y) = f(y/x), what is:

x (dz/dx) + y (dz/dy)?

 

[SpecialEd]

the expression x(dz/dx)+y(dz/dy) is exactly zero.

proof:

Let u=y/x

By the chain rule, we have dz/dx=f'(u)*(-y/x^2) and dz/dy=f'(u)*(1/x).

The terms in the expression above then cancel.