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Derivative can be added to the formula to react with gain only as a PD (proportional, derivative) controller or it can be added with the PI (proportional, integral) controller to make a full three-mode controller called PID (proportional, integral, and derivative). The derivative has the inverse effect of integral. Instead of specifying the time it takes to repeat the action caused by gain, the derivative is specified as the number of repeats in one minute. Notice that for derivative action, time is kept constant and the number of repeats that you want to occur will be changed. (Remember: for integral action, the F variable is time and the single repeat is the constant.) This means that if you want the controller response to become faster, you'd specify a larger number for the derivative, which would be more repeats per minute.
E.g., we can use the lawn-mowing example again to compare the action caused by the integral to the action caused by the derivative. We will show the results graphically in the Figure below. In the original lawn-mowing example it took 1 minute to make the first pass, which we called the amount of change caused by gain. If the derivative was set to a 2, it would mean that after gain completed the first pass, we would want the controller to increase the output so that two passes could be made during the next minute. If the derivative was specified to be 0.5, the response rate would have been slowed so that only one-half pass would be mowed during the next minute, and it would take 2 full minutes to mow the second pass.
Notice that a derivative of 2 would provide a similar effect as an integral of 0.5, and a derivative of 0.5 would be similar to an integral of 2. When you think of an integral, you must think in terms of minutes per repeat, and when you think of a derivative you must think in terms of repeats per minute. It should be noted that technicians generally only need to understand the effects of increasing and decreasing gain, reset, and rate, and they don't necessarily need to know how to do the actual calculations.
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