Newton's first law can be derived from his second law. Then why did he put forth the first law, when it's already contained in second law?
My teacher was saying that first law is valid for an inertial reference frame, while second law is also valid for non-inertial reference frame. Is that correct? (I don't think so, as I knew Newton's laws are invalid in non-inertial reference frame.)
You have to understand that inertia was a concept that was not well understood in Newton's time. People implicitly assumed that a body in motion would slow down and eventually stop after some time without the need of external force. (They assumed this because friction was not identified as a kind of push/pull force that surfaces in relative motion exert on each other but as an intrinsic property of every body.) When Newton deduced the first law it was a break from the past - a completely new way of thinking - and that is why (probably) he gave it a law of its own.
You are correct, the first law is not valid in a non-inertial frame. Think of yourself accelerating in a vehicle and looking at a rock on the street. The rock is accelerating in the opposite direction (in your eyes) and yet there is not external force acting on it.
Newton discusses in Principia Mathematica that force causes acceleration. However, without the first law it isn't necessary true that
$$F=0 \implies a=0$$
A person could get another notion of things that cause acceleration.
If you were Newton, you wouldn't be concerned with inertial reference frames as much as Aristotle's view on objects.
Anyhow, it is important to say that forces not only cause acceleration, they are the only thing that cause it.
Savvy?
