Suppose that the quantity y is a function of the quantity x, so that y = f(x). If f is differentiable, the rate of change of y with respect to x is the derivative dy/dx or f′(x). The rate of change is often with respect to time t. Suppose, now, that x denotes the displacement of a particle, at time t, on a directed line with origin O. Then the velocity is dx/dt or $\dot{x}$, the rate of change of x with respect to t, and the acceleration is d²x/dt² or $\ddot{x}$, the rate of change of the velocity with respect to t.

When the motion is in two or three dimensions, say

denoting the position vector of a particle, the velocity of the particle is the vector

and the acceleration is