GoldExI : FAQ : Derivative

Question

What is the difference between simple derivative (Math | Calculus| Simple dy/dx) and multi-point derivative (Math | Calculus| Multi-point dy/dx)?

Answer

The simple dy/dx works using the formulas

y_new_i = (y_{i+1}-y_{i})/(x_{i+1}-x_{i})

x_new_i = (x_{i+1}+x_{i})/2

Clearly, if the there are consecutive points with the same x-coordinates, this will result in division by 0. To avoid this, before applying the above algorithms, GoldExI fixes the curves. It searches for consecutive points with the same x-coordinates. If such a series of points is found, GoldExI substitutes this series of 2 or more points with one point with the same x-coordinate. The y-coordinate is the average of all y-coordinates in the series.

Multi-point algorithm assumes that point are equidistant in x! Then it just uses high-order formula to find derivative by N points. The number of points can be set in Options | Math. There you can also set what to do close to the edges of the curve. Since interpolation formula requires N points, it is not clear what to do close to the edges. You have the following options:

to skip calculation of derivative close to the edges. In this case the resulting curve will contain N points less than the original curve.
to calculate close to the edge assuming that missing points have the same y-coordinate as the first/last point of the curve.

In multi-point algorithm you can also correct the sign of the derivative for the curve direction. For example, the curve going from right to left has negative dx. Should the derivative be dy/dx or dy/|dx|? You can choose this in Options | Math.

General remarks

Digital differentiation often produces very noisy results just because of the small noise in the input data. Therefore it is strongly recommended that you measure the derivative, e.g. using lock-in technique. In the cases when it is not possible, smoothing the data after taking a derivative may help. Smooth the data using Math | Smooth. Multi-point dy/dx routine, is doing something like this intrinsically.