derivation of the distance formula?

Distance between points (x1,y1) and (x2,y2)
d ² = (x2 - x1) ² + (y2 - y1) ²
d = √ [ (x2 - x1) ² + (y2 - y1) ² ]

You mean physics classic dynamics distance or math distance formula?

Distance=Speed x Time

a = dv/dt
dv = a dt
so, v - v[0] = at
v = v[0] + at --------------[1]
v = dS/dt
dS = v dt
dS = (v[0] + at) dt
S = v[0]t + 1/2 a t^2 ----{2]

S = vt

speed=distance/time

distance=time*speed

time=speed*distance