implicit derivatives on a surface

Question

Given a surface defined implicitly by a function like $f(x,y,z) = c$, I'm trying to show that $\frac{dx}{dy}\frac{dy}{dz}\frac{dz}{dx} = -1 $ for derivatives taken along the surface. I have no clue where to start on this, it seems like by the chain rule it ought to be 1.