The Units Shift

Taking a derivative quietly changes the units, and the units tell you what you are measuring. Track a car's position in meters. Its derivative, velocity, comes out in meters per second. Take the derivative again and acceleration arrives in meters per second squared. Each step divides by another second of time.

This is a handy sanity check. If you compute a rate and the units look wrong — say, dollars where you expected dollars per day — you have made a mistake somewhere. The units are a built-in receipt for every derivative you take.