But this is something commonly used in physics.
If f(t) is the position function, then the speed function is . And the velocity (which means speed at an instant) at time t=a is
You can view it this way :
- speed is a ratio distance/time
- the derivative is a ratio "function displacement"/"units"
A formal way to view the velocity is to see that velocity at a time t=a is the difference of distance that occurred instantaneously. More basically, it is the difference of distance that occurred within a very very very little interval of time.
Let be this interval of time.
denotes the difference of distance within this very little time (around t=a). Dividing by will make the speed.
Because we want this interval to be very small, like null, we'll make
So finally, we're looking for , by definition.
Is it clear ?
NB : the acceleration function is