  In the context of the analysis of measured data, one is often faced with the
task to differentiate data numerically. Typically, this occurs when measured
data are concerned or data are evaluated numerically during the evolution of
partial or ordinary differential equations. Usually, one does not take care for
accuracy of the resulting estimates of derivatives because modern computers are
assumed to be accurate to many digits. But measurements yield intrinsic errors,
which are often much less accurate than the limit of the machine used, and
there exists the effect of ``loss of significance'', well known in numerical
mathematics and computational physics. The problem occurs primarily in
numerical subtraction, and clearly, the estimation of derivatives involves the
approximation of differences. In this article, we discuss several techniques
for the estimation of derivatives. As a novel aspect, we divide into local and
global methods, and explain the respective shortcomings. We have developed a
general scheme for global methods, and illustrate our ideas by spline smoothing
and spectral smoothing. The results from these less known techniques are
confronted with the ones from local methods. As typical for the latter, we
chose Savitzky-Golay filtering and finite differences. Two basic quantities are
used for characterization of results: The variance of the difference of the
true derivative and its estimate, and as important new characteristic, the
smoothness of the estimate. We apply the different techniques to numerically
produced data and demonstrate the application to data from an aeroacoustic
experiment. As a result, we find that global methods are generally preferable
if a smooth process is considered. For rough estimates local methods work
acceptably well.
