So we were asked that if $F(x,y,z)=0$,
show $\frac{\partial z}{\partial y}|_x \frac{\partial y}{\partial x}|_z \frac{\partial x}{\partial z} |_y = -1$.
Now I had to go about and show that $\frac{\partial z}{\partial y}|_x = -\frac{dz}{dy}$ etc, then substitute the partial derivates for their derivative equivalents and then I had the same thing as before but with standard derivatives instead of partials, and 3 negative signs. I then simplified the expression as if they were a bunch of fractions and the 3 negatives gave me a negative overall so I ended up with -1 as the answer.
But why can one not just treat partial derivates like fractions the same way you can for standard derivatives.
(I study Physics so I've been used to treating derivates as fractions for a long time)
On your surface, the differentials of $x$,$y$ and $z$ are related by $$ \frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}dz=0. $$ If you consider $z$ as an (implicitly defined) function of $x,y$, to find $\frac{\partial z}{\partial y}$ you make $dx=0$ and solve to get $$ \frac{\partial z}{\partial y}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}. $$ You can do the same for the other two. Multiply your results together and things cancel nicely to give $-1$.