This applies to all aircraft, big and small.
...except that your discourse did not quite answer the OP's question.
The reason that explicit rudder input is not used in modern airliners is, as TotalBeginner noted above, that the yaw damper systems also act as turn coordinators. A second reason is that the combination of spoiler and aileron for roll control negates the adverse yaw that many aileron-only airplanes suffer.
However, MANY (maybe even "most") small airplanes require some amount of rudder input along with aileron to coordinate the turn. In fact, aircraft with long wingspans (e.g., many gliders), turn better with rudder input in advance of the aileron input. This is, again, because the adverse yaw caused by the downward-deflected aileron (more lift => more drag on that side) will otherwise cause an initial yaw in the wrong direction.