Answer
$$ - 3{t^2}\sin t + 6t\cos t + \frac{1}{{2\sqrt t }}\sin 2t + 2\sqrt t \cos 2t$$
Work Step by Step
$$\eqalign{
& \frac{d}{{dt}}\left( {\left( {3{t^2}{\bf{i}} + \sqrt t {\bf{j}} - 2{t^{ - 1}}{\bf{k}}} \right) \cdot \left( {\cos t{\bf{i}} + \sin 2t{\bf{j}} - 3t{\bf{k}}} \right)} \right) \cr
& {\text{Calculate the dot product}} \cr
& \frac{d}{{dt}}\left( {3{t^2}\cos t + \sqrt t \sin 2t + 6} \right) \cr
& {\text{Calculate the derivative}} \cr
& = - 3{t^2}\sin t + 6t\cos t + \frac{1}{{2\sqrt t }}\sin 2t + 2\sqrt t \cos 2t \cr} $$
