I forgot to mention that I would also assume definition of electric field ##F=qE##
Anyway there is similar derivation starting with ##F=qE## (definition of electric field) and special relativity. That force is transformed to another frame of reference and it is shown that there is velocity...
I am working on derivation of Lorentz force. (I know that Lorentz force is in some sense definition of fields, but still there is nontrivial dependence on velocity).
I want to derive that the force is linear in components of velocity, so for example $$F_x=q(E+Av_x + Bv_y + Cv_z ),$$where ##A...