Looking over a selectioon of my books, you are right Mark, the dot notaion is more used for time than the prime, which is used for many other things as well.
It is, perhaps, not suprising that the Europeans should introduce the prime notation since most of their alphabets contain accented letters anyway.
On another note, Tyler, did you understand my comment about integrals which was directly addressed to one of your queries/difficulties?
I'm finding it hard to understand what exactly is means to have a differential after an expression ie "2xdx."

I am trying to supplement, not supplant, the excellent job Mark is doing at explanation here, but I agree with AZ that you have some of the basics muddled up.
Continuing this make sure you see the difference between
[tex]\frac{{{d^2}y}}{{d{x^2}}}\quad and\quad {\left( {\frac{{dy}}{{dx}}} \right)^2}[/tex]
In prime terms this is the difference between y'' and (y')
^{2}
go well