Introduction

Suppose a car travels from Mumbai to Pune. At every instant its position is being displaced from the starting point (Mumbai). The average speed over the whole journey is

but the car's speed is rarely constant — it is different at different moments. The speed at a particular instant is obtained by looking at a very small distance travelled in a very small time interval near that instant, forming the ratio (small distance)/(small time), and then letting that time interval shrink toward zero. The limiting value of this ratio is the car's speed at that instant.

This process — finding the limit of a ratio of a small change in one quantity to a small change in another, as the second change shrinks to zero — is exactly what differentiation means, and the limiting value itself is called the derivative. Applied to distance and time, it measures how quickly the car's position changes with time, i.e. speed is the rate of change of distance with time.

Velocity is speed together with a direction of motion. When the motion never reverses direction, the words 'speed' and 'velocity' can be used interchangeably without changing the answer to a problem.

More generally, for any function (not just distance-vs-time): the rate of change of a function at a point, with respect to its variable, is called the derivative of the function at that point, and the process of finding it is called differentiation.