The instantaneous velocity of an object is the velocity at a particular instant in time and is a vector quantity. It can be defined by considering the average velocity of a body. If a body moves a distance in a time , then its average velocity over is:
As velocity is displacement divided by time. As the time interval becomes smaller and smaller, this average velocity approximates better and better the velocity of the object at which the time interval is centred. To get the exact velocity at some time t, the limit of going to 0 is taken:
In this limit, the instantaneous velocity becomes the derivative of the displacement. In a similar way, the instantaneous speed of an object is the derivative of the object's distance and as such is the gradient of a distance-time graph.