Product rules apply.
(a) $\frac{d}{dt} \Big(a \vec A\Big) = \frac{da}{dt} \vec A + a \frac{d \vec A}{dt}$.
$\m$
(b) $\frac{d}{dt} \Big( \vec A \cdot \vec B\Big) = \vec A \cdot \frac{d\vec B}{dt} + \frac{d \vec A}{dt} \cdot \vec B$.
$\m$ (c) $\frac{d}{dt} \Big( \vec A \times \vec B\Big) = \vec A \times \frac{d\vec B}{dt} + \frac{d \vec A}{dt} \times \vec B$.