[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
therefore, v.dt = r d
(v.dt)/r = d
by definition of change,
in vector form dv = v[sub]f[/sub] - v[sub]i[/sub]