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# 10.6. Numerical Differentiation

## Problem

You have a table of experimentally obtained data representing some function and you'd like to approximate the derivatives of that function.

## Discussion

A well-respected professor once told me that numerical differentiation is death. That's a pretty strong statement, and what he meant was that once you start taking finite differences (a way to approximate derivatives numerically), accuracy goes downhill fast, ruining your results. This is because numerical differentiation can be very inaccurate due to its high sensitivity to inaccuracies in the values of the function being differentiated. This is in contrast to numerical integration, which is far more insensitive to functional inaccuracies because it has a smoothing effect that diminishes the effect of inaccuracies in the values of the function being integrated.

If you have an analytic form of the function under consideration, then you're far better off deriving an analytic expression for the derivative, as opposed to resorting to numerical differentiation. On the other hand, if you have experimentally obtained data with no analytic expression for the function, then you may have to resort to numerical differentiation. This latter case is the one we'll consider here.

By definition, the derivative of a function is:

When given tabulated data ...

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