Answer
$\dfrac{d(u \pm v)}{dt}=\dfrac{d}{dt}u \pm \dfrac{d}{dt}v $
Work Step by Step
Consider, $u=\lt u_x(t), u_y(t), u_z(t) \gt$ and $v=\lt v_x(t), v_y(t), v_z(t) \gt$
Apply product rule to get $\dfrac{d(u \pm v)}{dt}=\lt \dfrac{d}{dt}(u_xt \pm v_xt), \dfrac{d}{dt}(u_yt \pm v_yt),\dfrac{d}{dt}(u_zt \pm v_zt)$
or, $=\lt \dfrac{d}{dt}u_xt , \dfrac{d}{dt}u_y t,\dfrac{d}{dt}u_z t\gt \pm \lt \dfrac{d}{dt}v_xt , \dfrac{d}{dt}v_y t,\dfrac{d}{dt}v_z t\gt $
or, $= \dfrac{d}{dt}u \pm \dfrac{d}{dt}v $
Hence, $\dfrac{d(u \pm v)}{dt}=\dfrac{d}{dt}u \pm \dfrac{d}{dt}v $
