Partial Derivative f(x,y')=1: Why & True?

[MHD93]

let f(x, y') = x + y'
where y' = dy/dx
then is it true, and why, that the partial derivative of f with respect to y' = 1
in this case we consder dx/dy' = 0, as if they are independent of each other.

 

[obafgkmrns]

It's true. You can find an explicit example in the "Examples" section of http://en.wikipedia.org/wiki/Lagrangian_mechanics

 

[MHD93]

but since y' = dy/dx , then y depends on x, and y' might depend on x too
therefore there might be a relation between x and y', and dx/dy' doesn't necessarily equal zero

 

[obafgkmrns]

To tell the truth, that bothered me as well when I first encountered the concept. The professor was confused as to why I even asked the question. Consider a function f(x,y). Is there any problem with defining partial derivatives with respect to x and y? What if, unbeknownst to you, y is actually a function of x? Well, the answer is no problem. As far as f(x,y) is concerned, y is just a parameter that can take any value. You can substitute the relationship y(x) into the mix after you take the partial derivative.