Answer
$$0$$
Work Step by Step
$$\eqalign{
& {\text{Let }}{\bf{u}}\left( t \right) = \left\langle {1,t,{t^2}} \right\rangle {\text{ and }}{\bf{v}}\left( t \right) = \left\langle {{t^2}, - 2t,1} \right\rangle \cr
& {\text{Calculate }}{\bf{u}}\left( t \right) \cdot {\bf{v}}\left( t \right){\text{ using the dot product rule}} \cr
& \frac{d}{{dt}}\left[ {{\bf{u}}\left( t \right) \cdot {\bf{v}}\left( t \right)} \right] = {\bf{u}}\left( t \right) \cdot {\bf{v}}'\left( t \right) + {\bf{u}}'\left( t \right) \cdot {\bf{v}}\left( t \right) \cr
& = \left\langle {1,t,{t^2}} \right\rangle \cdot \left\langle {2t, - 2,0} \right\rangle + \left\langle {0,1,2t} \right\rangle \cdot \left\langle {{t^2}, - 2t,1} \right\rangle \cr
& = 2t - 2t + 0 + 0 - 2t + 2t \cr
& = 0 \cr} $$
