EDIT: Ok, your question was edited.
Try to look the "path" of the variables of your function.
By example, here we have:
$~F \to u, v, x, y~$ 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 "$~x~$".
$~f \to x ,~ y , ~u(x,y),~ v(x,y),~$ so you have to derivate everywhere $~x~$ is.
So, you'll have:
$$~\frac{df}{dx} + \frac{df}{du}\cdot \frac{du}{dx} + \frac{df}{dv}\cdot\frac{dv}{dx}~$$ where $~d~$ is the partial derivative.
I might be not clear as English isn't my mother-tongue, but I suggest you this link:
http://www.math.hmc.edu/calculus/tutorials/multichainrule/