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 ẋ, the rate of change of x with respect to t, and the acceleration is d2x/dt2 or ẍ, the rate of change of the velocity with respect to t.
In the reference to velocity and acceleration in the preceding paragraph, a common convention has been followed, in which the unit vector i in the positive direction along the line has been suppressed. Velocity and acceleration are in fact vector quantities, and in the 1-dimensional case above are equal to ẋi and ẍi. When the motion is in 2 or 3 dimensions, vectors are used explicitly. If r is the position vector of a particle, the velocity of the particle is the vector r̈̇ and the acceleration is r̈.