The speed of an object at any given instant is the rate at which it is changing its position. It can be defined mathematically as the first derivative of the position function with respect to time. In simpler terms, it is the limit of the average speed over an infinitesimal time interval.

If we let \(s(t)\) represent the position of the object as a function of time, then its speed \(v(t)\) at time \(t\) is given by:

$$v(t) = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$$

This definition of speed is valid for objects moving in one dimension. For objects moving in two or three dimensions, the speed is defined as the magnitude of the velocity vector, which represents both the speed and direction of the object's motion.