For purposes of illustration, let’s say we have a set of data, (x1,y1), (x2,y2), (x3, y3), (x4,y4), ...
A “two-point” derivative (span = 2) would use consecutive points for evaluation. For instance, the derivative’s set of data would be:
Note that when we calculate the independent values of the derivative, we get the midpoints of the original data set. Suppose we sampled for one second at a sampling rate of 10 Hz; the data set would have independent values of 0.00, 0.10, 0.20, 0.30, ... , 0.90, 1.00. But the derivative would have independent values of 0.05, 0.15, 0.25,...,0.95. Putting the original data in a table’s 1st column and its derivative in the 2nd column, we would see that the data is staggered.
The above algorithm is use for the derivative function within the calculator of PASCO Capstone and SPARKvue.
For sensors that use the derivative for calculating derived units (e.g. velocity and acceleration within the motion sensor), the following method is used.
The following shows the actual calculations when using a 3-point span, they are easily generalized to higher numbers of points. These increasing odd numbers (up to 13) are available within the Derived Velocity and Acceration Options within the Recording Conditions button within Capstone.
( y(t+1) + y(t) ) / 2 - ( y(t) + y(t-1) ) /2
Output Y= ______________________________
x(t+1) - x(t-1)
Output X = X(t+1) - X(t-1) / 2