It is the limit of average angular acceleration as $\Delta \mathrm{t}$ approaches zero, i.e.,

$\vec{\alpha}=\lim _{\Delta t \rightarrow 0} \frac{\Delta \vec{\omega}}{\Delta t}=\frac{\mathrm{d} \vec{\omega}}{\mathrm{dt}}$

since $\vec{\omega}=\frac{\mathrm{d} \vec{\theta}}{\mathrm{dt}}, \quad \therefore \vec{\alpha}=\frac{\mathrm{d} \vec{\omega}}{\mathrm{dt}}=\frac{\mathrm{d}^{2} \vec{\theta}}{\mathrm{dt}^{2}}, \quad$

Also$\quad \vec{\alpha}=\omega \frac{\mathrm{d} \vec{\omega}}{\mathrm{d} \theta}$