Click/tap in the right/left halves of the circle to cycle through time interval values.

Imagine a particle tied to the end of a string going around a circular path on a smooth horizontal surface with a uniform speed. You can work out the speed of the particle by knowing the time taken for one revolution and using speed is distance/time taken. Nothing big.

If you imagine what would happen if the string breaks you would come up with an answer- The particle travels along a straight line with a constant velocity. Since the direction of velocity did not change after the string broke, the velocity for the straight line motion is same as the velocity at the moment the string broke and so is directed along the tangent at the given moment. Again nothing big! The instantaneous velocity of a body moving along any curved path is of course along a tangent and its magnitude is equal to the speed of travel. You knew that.

But how do you get this instantaneous velocity from the definition of the velocity ?

Since velocity is displacement divided by time, you would need to look at two positions at two different instants to evaluate that. That would in fact give you average velocity- not the instantaneous velocity, for that you should work a little harder.

Suppose we did this-consider two instants so very close to each other, that we could not almost not tell them apart. That is the gap between the instants is real small. If we could get the positions of the moving particle at these two very close instants, we would almost get the instantaneous velocity. This would of course be an approximation to the instantaneous velocity, but a very good one. By really looking at two very close instants we would see the value of velocity calculated from the formula getting closer and closer to some value, which could be taken as very good approximation to instantaneous velocity.

In the animation we are looking at the positions of a moving particle at two different instants. the red vector is the displacement vector and the yellow vector is the average velocity vector. As we reduce the interval between the instants, the displacement vector gets shorter. As the denominator decreases, the numerator also decreases and the magnitude of average velocity approaches a specific value, and it's direction approaches a tangent. We can imagine that if the duration is zero the velocity at an instant is along the tangent. (Keep in mind that what seems obvious this way is not really that- how come we worked out the velocity when the duration is zero and the displacement is zero; We can not do that and we really have not done that. We are only saying that it can be approximated to something very reasonably close to what we physically perceive and know to be true.)