Explain the differences between average and instantaneous velocity with examples.

Hint: We will start with a simple definition of the velocity which is the time rate of change of the position of an object concerning the frame of reference. In a simple term velocity is a function of time. Velocity is also defined as the ratio of displacement and time.
Formula used:
$ \Rightarrow {\vec v_{average}} = \dfrac{{{{\vec S}_2} - {{\vec S}_1}}}{{{t_2} - {t_1}}}$
$ \Rightarrow {\vec v_{inst}} = {\lim _{\Delta t \to 0}}\dfrac{{{{\vec S}_2} - {{\vec S}_1}}}{{{t_2} - {t_1}}}$

Complete Step by step solution
Starting with the average velocity which is a ratio of the change in the displacement and time interval.
If $\Delta \vec S = {\vec S_2} - {\vec S_1}$is a change in position of an object and $\Delta t = {t_2} - {t_1}$is the time interval then average velocity will be
$ \Rightarrow {\vec v_{average}} = \dfrac{{{{\vec S}_2} - {{\vec S}_1}}}{{{t_2} - {t_1}}}$
\[ \Rightarrow {\vec v_{average}} = \dfrac{{\Delta \vec S}}{{\Delta t}}\]
For example: If a car takes a total of 3 hours to cover \[30{\text{ }}km\]distance towards the east direction and he travels back in the west direction about \[30{\text{ }}km\] then the average speed of the can be given as
$ \Rightarrow {\vec v_{average}} = \dfrac{{{{\vec S}_2} - {{\vec S}_1}}}{{{t_2} - {t_1}}}$
$ \Rightarrow {\vec v_{average}} = \dfrac{{30 - ( - 30)km}}{{3hrs}}$
$ \Rightarrow {\vec v_{average}} = \dfrac{{60km}}{{3hrs}}$
$\therefore {\vec v_{average}} = 20\dfrac{{km}}{{hr}}$
Now the instantaneous velocity can be defined as the velocity of an object at any instant of time or we can say that at any point on the path it is moving.
$ \Rightarrow {\vec v_{inst}} = {\lim _{\Delta t \to 0}}\dfrac{{{{\vec S}_2} - {{\vec S}_1}}}{{{t_2} - {t_1}}}$
$ \Rightarrow {\vec v_{inst}} = {\lim _{\Delta t \to 0}}\dfrac{{\Delta \vec S}}{{\Delta t}}$
Now as the limits exist it gives derivatives
$ \Rightarrow {\vec v_{inst}} = \dfrac{{d\vec S}}{{dt}}$
Hence the instantaneous velocity can be given as the displacement as the function of derivative of time
For example, A boy is riding a bike and he suddenly checks the velocity of a bike and at that instant of time, he found it be to$30\dfrac{{km}}{h}$. so that will be his instantaneous velocity.

Note: While dealing with velocity and displacement one should always remember to put an arrow on the respective quantity because as we know that velocity and displacement are vector quantities that require magnitude as well as direction also. Hence the arrow is used to show vector quantities.