Implicit functions equality

Question

Suppose that $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is such that $f(x,y,z) = 0$ for some $x,y,z$. Then, I am asked to show that, if each variable $x,y,z$ can be defined as an implicit function of the other ones, then $\frac{dx}{dy} \frac{dy}{dz} \frac{dz}{dx} = -1$. I tried to use the Implicit Function Theorem, but I failed at trying to obtain this equality. I would be grateful if someone could give me just a hint, not the entire answer since I would like to conclude it by myself unless I say the opposite.