Newtonian Mechanics cannot be applied in non-inertial frame of reference then how can we apply formula for normal force in an accelerating elevator? In my physics book there is a diagram in which a block is placed on a table. The table is exerting an upward force $ F_n $ (normal force) on the block and there is a downward force $ F_g $ (gravitational force) on the block
The equation for normal force is derived in this way:
$ F_{net,y} = MA_y $ (where $ M $ is mass and $ A_y $ is acceleration in vertical direction)
For the block we can write Newton's second law for positive upward y axis $ (F_{net,y} = MA_y) $ as $ F_n-F_g = MA_y $
Substituting $ MG $ for $ F_g $ (where $ M $ is mass and $ G $ is gravitational acceleration) we get $ F_n-MG=MA_y $
Then magnitude of normal force is $ F_n=MG+MA_y=M(G+A_y) $ for any vertical acceleration $ A_y $ of table and block (they might be in an accelerating elevator). 
But accelerating elevator would be non inertial frame and Newtonian Mechanics cannot be applied in non inertial frame of reference then how can we apply formula for normal force in an accelerating elevator? Even if the elevator is accelerating then it is not accelerating because of these forces so how can we apply this equation? 
 A: Looks like in your calculations you are NOT using non-inertial frames of reference at all.
If I understood you correctly you frame of reference is at rest. In this frame of reference the elevator (and the body inside the elevator) are moving with some acceleration. But your frame of reference is not! You can use usual mechanic formulas in this case.
But it was possible to attach the frame of reference to the elevator. Then things would be quite different.
In this frame of reference the coordinates of the body do not change over time. The body is not moving at all (remember, we are in a frame of reference attached to the elevator!). Our frame of reference is not inertial and you can't use usual formulas.
Lt's try and see what would happen.
There are still two forces applied to the body. One is gravitational force, the other one is the force exerted by the table. Magnitudes of these forces are different, total of these two forces can't be zero, so the body must accelerate. But it is not. It is not moving at all, it's coordinates are constant over time.
We have a contradiction, and this is because we tried to use usual mechanical formulas in non-inertial frame of reference.
Sometimes it is very convenient to use non-inertial frames of reference, and it is possible to use them, but mechanical formulas look different in these cases (you need to introduce some 'fictional' forces).
A: If you observe the elevator from the ground, you are evaluating its motion using an inertial reference frame (neglecting effect's of the earths rotation and revolution), and if these forces differ the elevator accelerates.  The forces are: gravity and the force moving the elevator.  If the elevator is accelerating relative to the ground, but you  choose to observe the elevator sitting on the elevator, you are using a non-inertial reference frame, and you must include the "fictitious" reaction force in that frame in addition to gravity and the force moving the elevator to keep the elevator fixed to an observer (you) moving with the elevator.
A: You can apply Newton's laws in any frame, they just won't give you the right results. To get the right results in non-inertial frames you need to add some fictitious forces. When elevator is accelerating upwards with acceleration $a$, any body lying in the elevator would feel $g_{effective}=g+a$. You won't be able to explain(in elevator's frame) why you feel heavy inside the elevator accelerating upwards unless you add a fictitious force ($ma$, here).
Basically fictitious forces are added to explain any peculiarity in the non-inertial frame. Feeling heavier? there must be a fictitious force weighing you down. Feeling lighter? there must be fictitious force acting opposite to gravity (elevator accelerating downwards). Feeling a radial outwards force? Must be the centrifugal force (yep, also fictitious). The ball you threw in the air didn't follow a parabolic trajectory? the Coriolis force(wouldn't you know it, fictitious) must be at work. So, when you're in a non-inertial frame, you explain all the peculiarities with some fictional force and that fictional force is taken into account to write newton's laws correctly in the non-inertial frame. Then only the results of newton's laws for a body will agree when looking from an inertial frame and a non inertial frame.
If you're outside the non-inertial frame  in some inertial frame, you can explain everything happening in the non inertial frame without using fictitious forces. The guy in the elevator feels heavier? That's because the elevator is accelerating him upwards!, so he feels pushed into the floor. But the guy in the non-inertial frame is ignorant about his situation, he thinks he's in a normal inertial frame. So, he explains the motion in his frame using these fictitious forces.
A: Individual forces doesn't depend on the frame of reference.
