[s]A typical answer to derive this is

Consider a particle in uniform circular motion with constant speed, v, and radius, r.
In a short time interval dt , the particle moves from A to B.
Distance traveled in time dt    =  v.dt

by definition of arc length
arc length  A B      =    r d\theta

therefore,  v.dt      =    r d\theta
              (v.dt)/r  =    d\theta

by definition of change,
  in vector form          dv = v[sub]f[/sub] - v[sub]i[/sub]