What is instantaneous acceleration? Explain with the help of a derivative.

Hint: To understand instantaneous acceleration we will first understand what acceleration means. The acceleration of a particle is defined as the rate of change of its velocity with respect to time. Since velocity is a vector quantity, acceleration is also a vector quantity. Now, any rate change measure can be classified into two types: average and instantaneous. We shall see the second one in detail.

Complete step-by-step answer:
The instantaneous acceleration of a particle, as the name suggests is the acceleration of the particle at a particular instant in time and not its acceleration over the particle’s entire course of motion.
Let us say, the particle is moving with a velocity which is denoted by $v\left( t \right)$. Here, the velocity function in time ‘t’ is changing with time. Then, the instantaneous acceleration of the particle at any time ‘${{t}_{1}}$ ’ will be given by:
$\Rightarrow {{a}_{inst}}={{\left. \dfrac{d\left[ v\left( t \right) \right]}{dt} \right|}_{t={{t}_{1}}}}$
This is the derivative form of instantaneous acceleration.
Hence, the instantaneous acceleration of a particle has been defined with the help of a derivative.

Additional Information: The average acceleration of a particle is defined as the total change in velocity over a period of time upon total time that has passed. It indicates the average change in speed with respect to time and not the changes in speed at every instant.
If the speed of the particle is $v\left( {{t}_{1}} \right)$ at an instant ‘${{t}_{1}}$ ’ and $v\left( {{t}_{2}} \right)$ at an instant ‘${{t}_{2}}$’, then the average acceleration of the particle is given by:
$\Rightarrow {{a}_{average}}=\dfrac{v\left( {{t}_{2}} \right)-v\left( {{t}_{1}} \right)}{{{t}_{2}}-{{t}_{1}}}$ .

Note: We saw the meaning and definition of instantaneous acceleration and average acceleration in the above section. The average and instantaneous accelerations of a particle could be equal if at all the instantaneous acceleration of the body is equal. That is, in the case of constant acceleration, the average and instantaneous acceleration of a particle is the same.