The instantaneous acceleration of a particle is its acceleration at a particular instant of time.

Instantaneous Acceleration is

$$ \vec{a}_{\mathrm{ins}}=\vec{a}=\lim \limits_{\Delta t \rightarrow 0} \frac{\Delta \vec{v}}{\Delta t}=\frac{d \vec{v}}{d t} $$

So, $\vec{a}=\frac{d \vec{v}}{d t}=\frac{d^2 \vec{x}}{d t^2}$