Ok...one last example that I have found on an MIT video. (The video is short and this is example is done in the first five minutes, it is here. They also try and relate it to a total derivative.
The example is:
Suppose Z = x^2 + y^2 {\rm and }\,\, x = u^2 - v^2 {\rm and }\,\, y= uv ...