How to derive Newton's third law …

Derivation of Newton’s third law of motion from Newton’s second law of motion

Consider an isolated system of two bodies A & B mutually interacting with each other, provided there is no external force acting on the system. 

Let FAB, be the force exerted on body B by body A and FBA be the force exerted by body B on A.

Suppose that due to these forces FAB and FBA, dp1/dt and dp2/dt be the rate of the change of momentum of these bodies respectively. 

Then, FBA = dp1/dt ---------- (i) 
=> FAB = dp2/dt---------- (ii) 

Adding equations (i) and (ii), we get, 
FBA + FAB = dp1/dt + dp2/dt
⇒ FBA + FAB = d(p1 + p2)/dt 

If no external force acts on the system, then 


d(p1 + p2)/dt = 0 
⇒ FBA + FAB = 0
⇒ FBA = - FAB---------- (iii) 

the above equation (iii) represents the Newton's third law of motion (i.e., for every action there is equal and opposite reaction).