Given that $u(3)=($:$1,0,-3$:$),u^{\text{'}}(3)=($:$6,1,1$:$),$ and $v(t)=($:$2t^2,t,2t+3$:$),$ find $r^{\text{'}}(3),$ where $r(t)=u(t)v(t).$

$r^{\text{'}}(3)=($:$,-74,$:$)$