I would like to prove the following identity, where $x,y,z$ are linked by a functional relation $f(x,y,z)=0$ and where the parentheses denote differentiation while keeping the indicated variable constant:
\begin{equation} \left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y=-1 \end{equation}
Can you help?