I refered these two questions
How to interpret instantaneous velocity using limit?
and I understood how instantaneous velocity is defined. But why do we define it? Velocity is rate of change of displacement.A change cannot occur at an instant.Alwasy an interval is neede for a change.Then why do we define instantaneous velocity? Is it just to study how the object may move in a very small interval arround an instant or is there something else?
Then why do we define instantaneous velocity?
We define instantaneous velocity because there are applications where we want to know what the speed of an object is at some instant (as you point out this is somewhat poor terminology since velocity denotes the change of position over a time interval, but it looks like you have a grasp of what we really mean by instantaneous, so I will use that term here). For example, the speedometer in your car tells you your instantaneous velocity while driving. Or in collisions if we want to know the momentum of each object right before a collision we would look at the instantaneous velocity.
I would argue that instantaneous velocity is all we really care about. Average velocity is subjectively dependent on the length of the time interval you are averaging over, so I would say at best average velocity is just a good tool for calculating other things of interest. In my experience you first learn about average velocity, and then it is rarely mentioned after that. Instantaneous velocity uses an interval that approaches $0$, so there is no subjective interval length.
From a comment on another answer: But my doubt is, if we multiply the time interval with the magnitude of average velocity, we get the distance traveled.
This is not true in general. The example of an object moving in a circle is a good one. If we define our time interval to be the time it takes for our object to move through one revolution around our circle, then the average velocity is actually $0$, but the object definitely traveled a non-zero distance around the circle. If we want the total distance traveled then we would actually want to integrate over the magnitude of the instantaneous velocity across the entire trip.
Limit is actually means the derivative for a continous and single valued functions. So by taking derivative of a $x(t)$ we are actually calculating the velocity with respect to any time $v(t)$.
Why we are taking the derivative, because we want to find the relationship between velocity of the object and time.
