This is just me being pedantic :)
I don't see anything wrong with it.
Acceptable notation for what?
That thought occurred to me as well.
Unclear. What is y ? What is y(x) ? What is y'(x)?
The apostrophe mark may mean either, "the other specified...", or it may mean, "first derivative of...".
If you are giving your function the variable name of y(x), then most likely one would use y'(x) to mean, the first derivative of y(x) with respect to x.
I don't see anything wrong with y'(x), but I guess it depends on whether you had something non-standard in mind.
I have never been a fan of calling y(x) a function. y is the function. y(x) is a number in the range of y. y' is the derivative of y. y'(x) is the value of y' at x, and also the derivative of y at x.
Separate names with a comma.