Let $f(u,v) = c$ where $u(x,y) , v(x,y)$ are functions and $c$ is constant. Can we conclude $\frac{\partial f}{\partial v} = \frac{\partial f}{\partial u} = 0$ ? It really sounds confusing to me but I've tried many examples and also the definition of partial derivative , and it was true ! What's the problem here ?
Main question : Suppose $f$ is a differentiable function . If $z$ is a differentiable function with respect to $x$ and $y$ and defined in $f(xz,yz) = 1$ prove that :$x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = -z$