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A motorbike starts from rest and moves in a straight line. Its position changes over time, described by a position function.

As the position changes with time, the slope of the plot gives velocity—the rate of change in position, which is the first derivative of position. Its magnitude is speed, as shown on the speedometer.

When velocity changes with time, either by speeding up or slowing down, this change is called acceleration—the second derivative of the position. A linear increase in velocity gives positive acceleration, and a decrease gives negative acceleration.

Acceleration can also vary. A quick twist of the throttle or sudden braking abruptly changes it. This rate of change of acceleration is called jerk —the third derivative of the position. Jerk describes how gradual or abrupt the acceleration changes.

These derivatives form a hierarchy: position gives location, velocity is the rate of position change, acceleration is the rate of velocity change, and jerk is the rate of acceleration change. In everyday life, motion is primarily described by velocity and acceleration.

But when highly precise cruise control is needed, higher-order derivatives like jerk are also considered.