How is the parital derivative (even in Leibniz notation) ambiguous?

[find_the_fun]

I had taken a multi-variable calculus course and since have misplaced my notes. I recall the prof inventing his own notation because somewhere partial derivatives using Leibniz notation don't show the correct path. I think it was something like if you had a function f(x,y)=z and y depended on x and t then if you write

\(\displaystyle \frac{\partial z}{\partial x}\) it's unclear which x is being referred to. Is this right? If no does anyone else know of an amibguity that arises?

 

[Fantini]

In Calculus on Manifolds, Spivak does mention the following:

$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial x}.$$

Note that $f$ has distinct meanings on each side. Another usual notation is $f_x$.