Independent variables of the Lagrangian

[JanEnClaesen]

Why are y and y' treated as independent variables, while they are not?

Another slightly related question:
if ' = d/dt then df'/dg' = df/dg because f' = df/dg g', but if we differentiate f' to g' we implicitly assume that df/dg is independent of g', is it?

 

[JanEnClaesen]

Perhaps because y and y' are the values of y and y' at the boundaries of a path, in which case they do seem independent.
Hence a natural follow-up question, instead of binding the positions, why don't we bind the velocities?

 

[UltrafastPED]

Here is a discussion of this same topic:
http://physics.stackexchange.com/questions/885/why-does-calculus-of-variations-work

Multiple answers are proposed - each convincingly.

The simplest one is that y and y' are independent algebraically.