EDIT: Ok, your question was edited.
Try to look the "path" of the variables of your function.
By example, here we have:
F -> u, v, x, y WHERE u and v depends of x and y.
If you define a new function, f, which has x,y as variables, it remains only two variables (in fact).
By example, if you want to determine the partial derivative of f, just "follow" the path to "x".
f -> x , y , u(x,y), v(x,y), so you have to derivate everywhere x is.
So, you'll have:
df/dx + (df/du)(du/dx) + (df/dv)(dv/dx) where d is the partial derivative.
I might be not clear as English isn't my mothertongue, but I suggest you this link: