Acceleration. Rectilinear motion

In general, $v$ will be a function of $t$. Now let $t$ take on an increment $\Delta t$, then $v$ takes on an increment $\Delta v$, and $\frac{\Delta v}{\Delta t}$ is the average acceleration of P during the time interval $\Delta t$. We define the acceleration $a$ at any instant as the limit of the ratio $\frac{\Delta v}{\Delta t}$ as $\Delta t$ approaches the limit zero; that is,

$$a = \lim_{\Delta t \to 0} \left ( \frac{\Delta v}{\Delta t} \right ),$$

or,

$$a = \frac{dv}{dt}$$ (6.28)

The acceleration is the derivative of the velocity with respect to time.