Question

derive F= ma with usual notation

Answer 1

W.K.T. K.dP/dT = F (Newton's 2nd law)

We know that dP = M.dV
=> F= k.M ×dV/dT
dV/dT is nothing but rate of change of velocity i.e, acceleration = a
k=1 experimentally
Hence,F=ma

Answer 2

It give by newton in a 2nd law of motion.
by a 2nd law of motion.
consider a body m moving with velocity v
this linear momentum of a body is given by: 
p=mv 
now, according to newton 2nd law
force is directly propotional to the rate of change of momentum that is
F α dp/dt
F  = k dp/dt
F = k d(mv)/dt
F = k md(v)/dt
F = k ma
Experimentally k =1
F = k ma