The average velocity tells us how fast an object has been moving over a given time interval but does not tell us how fast it moves at different instants of time during that interval. For this, we define instantaneous velocity.

The velocity at an instant is defined as the limit of the average velocity as the time interval $\Delta t$ becomes infinitesimally small.

In other words,

$v=\lim _{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}$

$v=\frac{d x}{d t}=\dot{x}$

In the language of calculus it is the differential coefficient of $x$ with respect to $t$ and is denoted by $\frac{d x}{d t}$.

We can obtain the value of velocity at an instant either graphically or numerically.

$(1)$ Graphical Method :

Suppose the below given $x \rightarrow t$ graph is for non-uniform motion of the car and we want to obtain graphically the value of velocity at time $t=4 \mathrm{~s}$.

For this case, the variation of velocity with time is found to be as shown in figure.

Suppose, by keeping $t=4 \mathrm{~s}$ in mind, if we take different time intervals $\Delta t_{1}, \Delta t_{2}, \Delta t_{3}, \ldots$ then for these time intervals displacement will be $\mathrm{P}_{1} \mathrm{P}_{2}, \mathrm{Q}_{1} \mathrm{Q}_{2}, \mathrm{~T}_{1} \mathrm{~T}_{2}, \ldots$ and corresponding displacement are $\Delta x_{1}, \Delta x_{2}, \Delta x_{3}, \ldots$