In complement to an old thread in the former Mathcad collab .
Don't mistake, "points derivative" is not obsolete, not at all !
In industrial control systems, measured data are accumulated
and pushed down in 'Z' accumulators [5 points is common].
From the 'Z' accumulator, at each loop scan, values are put
in the recursive algorithm and depending the construct of the
algorithm you apply, then comes out :
1.the derivative of the process measured variable [PV]
2. the integral
3.mostly the control algorithm [PID] as specified by
P [proportional], I [integral], D [derivative] modes.
The five points derivative does in fact processes only
four  points. The calculated derivative comes out
of the central position, i.e: the 5th point.
This type of derivative is called "T bar". There are very many
more ways to extract the point derivatives, but they all pertain
to numerical algorithms generally behind the scene .
For instance, the "cubic splines" before the tridiagonal solver
that is a lot easier to exploit like the built-in Mathcad .