  Data analysis for the LISA Technology package (LTP) experiment to be flown
aboard the LISA Pathfinder mission requires the solution of the system dynamics
for the calculation of the force acting on the test masses (TMs) starting from
interferometer position data. The need for a solution to this problem has
prompted us to implement a discrete time domain derivative estimator suited for
the LTP experiment requirements. We first report on the mathematical procedures
for the definition of two methods; the first based on a parabolic fit
approximation and the second based on a Taylor series expansion. These two
methods are then generalized and incorporated in a more general class of five
point discrete derivative estimators. The same procedure employed for the
second derivative can be applied to the estimation of the first derivative and
of a data smoother allowing defining a class of simple five points estimators
for both. The performances of three particular realization of the five point
second derivative estimator are analyzed with simulated noisy data. This
analysis pointed out that those estimators introducing large amount of high
frequency noise can determine systematic errors in the estimation of low
frequencies noise levels.
