Evaluate the derivative by using the appropriate Product Rule where $r_1(t)=($:$t^2,t^3,4t$:$),r(2)=($:$2,1,0$:$),$ and $r^{\text{'}}(2)=($:$1,4,3$:$).$

$($Give an exact answer. Use symbolic notation and fractions where needed.$)$

$\frac{d}{dt}(r(t)*r_1(t)){|}_t||=2=$