What is the second derivative of displacement?

The second derivative of displacement with respect to time is instantaneous acceleration: a = d²s/dt². It measures how quickly velocity changes at a particular moment.

The first derivative of displacement is velocity, v = ds/dt. Differentiating velocity once more gives acceleration, a = dv/dt. Combining those relationships produces the second-derivative formula. Its SI unit is metres per second squared (m/s²).

A common mix-up is confusing acceleration with average acceleration. Average acceleration is the change in velocity divided by a finite time interval, while instantaneous acceleration is defined using an infinitesimally small interval and is represented by the second time derivative. Another nearby concept is jerk, which is the third derivative of displacement and describes how acceleration changes.

In calculus, a second derivative also describes the curvature or concavity of a graph. For a displacement–time graph, its value indicates whether velocity is increasing or decreasing at that moment; a negative value means acceleration opposite the chosen positive direction.