Motion in One Dimension - Instantaneous velocity and acceleration

Sometimes it is not good enough to know the average velocity or acceleration and the instantaneous velocity is required. To do this, the time interval must be made increasingly close to zero. This corresponds to the limit of d/t as t goes to zero. This is the derivative of the magnitude of displacement with respect to time. Another way to look at this, is that the velocity is the change in displacement over time and by the informal definition of a derivative:

In this example x, v and a are function with respect to time.
The same is true of acceleration and velocity. See above.