Can the Newton's laws be derived from each other in a specific order only(2nd from 1st only and not from 3rd)? In my opinion we can derive Newton's laws in a specific order only that is 2nd from 1st and 3rd from 2nd and first only.
Let us suppose there is a body B which is in its initial state P(i). Now as per Newton's 1st law we have that to take the body to some state P(j) in time t(ij) we need to apply force on it.
From above we certainly know that to Change the state of a body force must be applied and hence we can define force F as F= G(P(ij)) or in unit time t(ij) we have F= f(P(ij)/t(ij))
 A: This is a common misconception that Newton's first law is unnecessary or that it can be derived from Newton's second law; ''If we put the force equal to zero in Newton's second law then we get the first hence the first law is redundant''. But this is wrong. What is a force? Newton's first law defines what a force is! And Newton's second law describes how this defined force acts on an object.
A: Our understanding of the logical structure of newtonian mechanics has benefitted from the shift to relativistic mechanics.
In terms of special relativity there is the concept of Minkowski spacetime. The geometric properties of Minkowski spacetime are expressed in the form of the Minkowski metric. The Minkowski metric expresses (among other things) a relation between space and time.
So:
With the benefit of hindsight we see that in order to have good logical structure in our theory of motion it is necessary to declare our concept of the geometric properties of the arena where the physics is taking place.
In effect, that is what the first law is doing. In effect the first law covers that newtonian space is assumed to have the same geometric properties as Euclidean space.
Also: in effect the first law expresses that an object in inertial motion will in equal intervals of time cover equal intervals of spatial distance; in that way a relation between space and time is expressed.

The second law is logically independent of the first law. The second law is expressed as a differential relation. Expressing a differential relation means that at each point in time you are looking what is happening at that specific point in time. That is, the evaluation is strictly local.
In the 18th century it was recognized that in parallel to Euclidean geometry there are two other spatial geometries that are just as self-consistent as Euclidean geometry is: hyperbolic geometry and spherical geometry.
In that sense we can recognize three spatial geometries: hyperbolic, Euclidean, spherical. The second law can be used in conjunction with each of those three geometries, without self-contradiction arising, because the evaluation that the second law expresses is a strictly local evaluation. That demonstrates that the second law is logically independent from the first law.
