The instantaneous velocity in calculus is the first derivative of a function that expresses distance as a function of time. The instantaneous velocity is also depicted graphically as the slope of the tangent line at a specific point.
The average velocity for any object is determined by the change in distance divided by the change in time for two points on the graph of distance versus time. If the speed is either accelerating or decelerating, however, the average velocity does not fully describe the motion. The instantaneous velocity is obtained by taking the limit as the difference between two time points goes to zero. As the difference between the two points approaches zero, the average velocity approaches the instantaneous velocity. This is referred to in calculus as the derivative or the slope of the tangent line at that point. The instantaneous velocity at each point in time gives a graph or function for the velocity versus time. Acceleration is then determined by taking the derivative of a function relating the instantaneous velocity to time.
Instantaneous velocity can be positive or negative depending on if the object in question is moving in the positive or negative direction. The instantaneous speed of an object is the absolute value of the instantaneous velocity.