Technically, the gradient is extrapolating data!!!
When you have a series of position measurements..... you require 2 position measurements to get to a velocity estimate, and require another position measurement to get to an acceleration measurement.
Assume Time = T1, T2, T3, ...
Position 1 = only know position P1 and T1
Position 2 = know P1 and P2 and T1 and T2
and estimate V1 = (P2 - P1) / (T2-T1) = average velocity getting from P1 to P2,
and corresponding time TV1 = (T2-T1)/2
Position 3 = know P1 and P2 and estimate of V1 (above) and P3 and V2 = (P3 - P2) / (T3 - T2)
and corresponding time TV2 = (T3-T2)/2
and know acceleration 1 A1 = (V2-V1)/(TV2 - TV1) =
((P3 - P2) / (T3 - T2) - (P2 - P1) / (T2-T1)) / ((T3-T2)/2 - (T2-T1)/2)