You have a very good understanding of this. is very different from though. The difference is the first expression is the same as f'(x) or y' and means "the derivative of function y with respect to x". It is not an operation on anything. The second expression I mentioned though is just that. It means take the derivative of whatever comes after d/dx with respect to x.
So again, the difference is between saying "this is the derivative of y" and "take the derivative of what's after this with respect to x".