Consider speed and its derivative, acceleration. For example, if you sped up from 60 to 80 km/h in five seconds, your rate of change - your acceleration - would be four kilometres per hour per second. If we wanted to see how well your bike gets from 0 to 100 km/h, it would be difficult to tell from speed data alone, and the picture would be far from complete. An acceleration chart, however, shows quickly and easily the rate of change of velocity, and puts a number on that acceleration that can be used for easy comparison. Likewise, if we wanted to see how smoothly you rode your bike on the freeway, we could look at your speed data and see how it varies over time; the less that variation, the smoother you are. Going through speed data alone to find those variations would require a lot of time and effort, but a quick look at acceleration data would show right away how quickly speed changes at any given moment.
The same math can be applied to any channel, not just speed: The rate of change of throttle position will show - and put numbers on - how smoothly a rider opens the throttle exiting a corner, or how quickly it is closed at the end of the straight. Another example is lean angle: The derivative of lean angle will show the motorcycle's roll rate, or how quickly it goes from full lean to full lean in a chicane. The rate of change of any data channel will show a completely new layer of information that can be used for analysis.
At a basic level, looking at raw data is fine for determining performance or for comparing bikes or riders to each other. But at a certain level, all that raw data starts to look the same; pro-level riders all get around the track at about the same speed, open the throttle at about the same time in every corner, and brake at about the same place for every corner. To find differences, it becomes necessary to delve deeper into the data, and that means more time is spent looking at those rate-of-change derivative channels.
Recently, Sport Rider published a profile of http://www.sportrider.com/features/146_1303_team_roberts_mike_sinclair/">Team Roberts' Mike Sinclair, who led the team's development of data acquisition in its early days. Sinclair talks of using the second derivative of raw data for analysis - the rate of change of the rate of change of a particular channel. If we are referring to lean angle, this would mean looking at not only how quickly the bike leans from side to side, but also how quickly it gets to that roll rate - an event that occurs in just a fraction of a second.
At the higher levels, that is what is required to show the necessary detail, and is an indication of the tiny slivers of time and data that teams in MotoGP and World Superbike are dealing with to find every last bit of performance.