# Derivative and Rate of Change

Figure 1: This chart shows data for a lap at Road Atlanta, with speed in black and the derivative of speed in red. If we were to overlay the longitudinal acceleration data channel here, it would be exactly the same as the speed derivative data.

Studying how a parameter changes over time is a large part of analyzing data, and can reveal plenty of information about the rider’s inputs along with the machine setup. For example, we are concerned not only with how hard the rider can brake, but also how quickly (or slowly) the rider applies the brakes at the beginning of the braking zone. A derivative channel of brake pressure or braking G will show that information quickly and easily.

We have already discussed derivatives briefly, in the Introduction to G-Forces and Acceleration section. Acceleration is defined as the rate of change of speed over time:

$a=\frac{\Delta v}{\Delta t}$

$a=\text{acceleration}\\ \Delta v=\text{change in velocity}\\ \Delta t=\text{change in time}$

The example used was: If your speed goes from zero to 60 mph in three seconds, you are accelerating at a rate of 20 mph per second. Derivatives come into play when we look at very small increments of time and how data changes over that time. In this example, if we look at the change in velocity over an increment of time that  is small enough – infinitesimally small – we can see the instantaneous acceleration at any given time. Figure 1 shows data for a lap at Road Atlanta, with speed at the top in black and the derivative of speed in red below. Note that as speed increases – acceleration – the derivative trace is positive, and as speed decreases – deceleration – the derivative trace is negative. The derivative trace is exactly the same as the longitudinal acceleration trace would show for this data, just as you would expect.

Figure 2: For the same lap at Road Atlanta, this data shows throttle position in blue and the derivative of throttle position in red. Note how the derivative trace changes to show various characteristics in the throttle position data. Here we can see how quickly the rider opens and closes the throttle, as well as how smoothly.

Figure 2 shows data for the same lap, with throttle position in blue and the derivative of throttle position in red. Here we can see how quickly the rider opens and closes the throttle; note the sharp negative spikes in the derivative channel when the throttle is closed, and the positive areas when the throttle is opened. Another benefit of the derivative channel is that it shows how smoothly the throttle is opened – a constant, positive value for the derivative trace shows the throttle being opened at a constant rate. In later sections we will show how the derivative channel can be used to study various other apects, including braking and turning.