Mathematics

Instantaneous velocity captures how fast and in which direction a particle is moving at one precise moment—not over an interval. The key insight is that it's the limit of average velocity as the time interval shrinks to zero, which is exactly what a derivative represents: $v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$. This definition lets you pin down motion at a single instant rather than describing it only over a span of time.

Instantaneous velocity captures how fast and in which direction a particle is moving at one precise moment—not over an interval. The key insight is that it's the limit of average velocity as the time interval shrinks to zero, which is exactly what a derivative represents: $v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$. This definition lets you pin down motion at a single instant rather than describing it only over a span of time.