Question

state the third law of motion with mathematical derivation  

Answer 1

For every action there is an equal and opposite reaction.  
In a system of isolated bodies, if a body applies a force F on the second body, then it in turn exerts a force -F on the first body.
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Derivation

Let there be two bodies A and B respectively. Let them be isolated from other bodies and external influences. Let their masses be m1 and m2 respectively. Let their velocities be u1 and u2 at time t1 and be v1 and v2 at time t2 respectively.

Let the external force on this system of two bodies is zero.  According to the second law of Newton. Force is the time rate of change of momentum. So when force is zero, the change in total linear momentum is zero.  Hence the total linear momentum of these two bodies is a constant in time.

So, m1 u1 + m2 u2 = m1 v1 + m2 v2
 
      m2 u2 - m2 v2 = m1 v1 - m1 u1
      m2 (u2 - v2 ) = m1 (v1 - u1)              now divide by (t2 - t1 )
      - m2 (v2 - u2) / (t2 - t1)   =  m1 (v1 - u1 )/ (t2 - t1 )
     - m2  a2 = m1 a1                as acceleration is change of velocity/time duration
     - F2 = F1                           as Force acting on a body = m a
  
 
   So, Force acting on body 1 is equal to the force acting on the body 2 in magnitude and opposite in direction.