how to derive newtons second formula of motion

Newton's 2nd law of motion states that rate of change of momentum is directly proportional to the applied force and the change takes place in the direction of applied.

So accordingly, F = k dP / dt           (here, k is the constant of proportionality)

Now, if a body of mass 'm' moving with initial velocity 'u'  changes to final velocity 'v' in given time 't',

then, change in momentum ( dP ) = m ( v - u )

and rate of change of momentum ( dP/ dt ) = m ( v - u ) / t

now following from definition, F = k dP / dt = m ( v - u ) / t

or,                                                 F  = K ma    [ (v-u)/ t = a ]

when m= 1 kg , a = 1 m/s² and F = 1 N,   

     K = 1

so, F = ma    ; which is the formula or mathematical form of Newton's 2nd law of motion.