"The time rate of change of velocity for corresponding interval of time is called average acceleration."

$\text { Average acceleration }=\frac{\text { Change in velocity }}{\text { Time interval }}$

The average acceleration $\vec{a}$ of an object for a time interval $\Delta t$ moving in $x y$-plane is the change

in velocity divided by the time interval :

$\vec{a}=\frac{\overrightarrow{\Delta v}}{\Delta t}=\frac{\Delta\left(v_{x} \hat{i}+v_{y} \hat{j}\right)}{\Delta t}=\frac{\Delta v_{x}}{\Delta t} \hat{i}+\frac{\Delta v_{y}}{\Delta t} \hat{j} \quad \vec{a}=a_{x} \hat{i}+a_{y} \hat{j}ac$

The acceleration (instantaneous acceleration) is the limiting value of the average acceleration as the time interval approaches zero.

$\left(\vec{a}=\lim _{\Delta t \rightarrow 0} \frac{\overrightarrow{\Delta v}}{\Delta t}=\frac{\overrightarrow{d v}}{d t}\right)$

Since $\overrightarrow{\Delta v}=\Delta v_{x} \hat{i}+\Delta v_{y} \hat{j}$, we have

$\vec{a}=\hat{i} \lim _{\Delta t \rightarrow 0} \frac{\Delta v_{x}}{\Delta t}+\hat{j} \lim _{\Delta t \rightarrow 0} \frac{\Delta v_{y}}{\Delta t}=\frac{d v_{x}}{d t} \hat{i}+\frac{d v_{y}}{d t} \hat{j}$

$\vec{a}=a_{x} \hat{i}+a_{y} \hat{j}$

where, $a_{x}=\frac{d v_{x}}{d t}, a_{y}=\frac{d v_{y}}{d t}$

Equation $(1)$ represent that differentiation of velocity w.r.t. time gives acceleration.

$\vec{a}=\frac{\overrightarrow{d v}}{d t}=\frac{d}{d t}\left(\frac{\overrightarrow{d r}}{d t}\right)=\frac{d^{2} \vec{r}}{d t^{2}}=\ddot{\vec{r}}$

Equation$ (3)$ represent that double differentiation of position (displacement) w.r.t. time gives acceleration.