Let's imagine you are walking and at each hour you note the distance you have traveled:
| 10 h 00 | 0 km |
| 11 h 00 | 1 km |
| 12 h 00 | 4 km |
| 13 h 00 | 9 km |
| 14 h 00 | 16 km |
| 15 h 00 | 25 km |
What was your average speed for the entire run?
$$\frac{25\ km - 0\ km}{15\ h - 10\ h} = 5\ km/h$$
What was your average speed at each hour?
At 11 h 00:
$$\frac{4\ km - 0\ km}{12\ h - 10\ h} = 2\ km/h$$
At 12 h 00:
$$\frac{9\ km - 1\ km}{13\ h - 11\ h} = 4\ km/h$$
At 13 h 00:
$$\frac{16\ km - 4\ km}{14\ h - 12\ h} = 6\ km/h$$
At 14 h 00:
$$\frac{25\ km - 9\ km}{15\ h - 13\ h} = 8\ km/h$$
As you can see, the equation has the form
$$v = \frac{\Delta position}{\Delta time}\ or\ v = \frac{\Delta x}{\Delta t}$$
If we had taken measurements more often, we could have been even more precise as to what was your speed at any point during your walk. Imagine taking measurements every second. Then the ##\Delta t## would have been 2 seconds instead of 2 hours giving you practically the instantaneous velocity at each point of your walk.
So the definition of the instantaneous velocity ##\frac{dx}{dt}## would be the one where ##dt## tends to ##0## or:
$$\frac{dx}{dt} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$$
The ##d## stands for "small difference" as opposed to ##\Delta## (greek letter for ##d##) which stands for "large difference".
In the following image, the velocities we calculated are represented by the slope of the yellow line, and the instantaneous velocity at ##4:00## is represented by the slope of the red line (which is tangent to the curve ##f##).
