If $u(t)=($:$sin(8t),cos(8t),t$:$)$ and $v(t)=($:$t,cos(8t),sin(8t)$:$),$ use the theorem given below to find $\frac{d}{dt}[u(t)v(t)].$

Suppose $u$ and $v$ are differentiable vector functions. Then $\frac{d}{dt}[u(t)v(t)]=u^{\text{'}}(t)v(t)+u(t)v^{\text{'}}(t).$

$($:$8tsin(8t)-cos(8t)+8sin(16t),2t-8sin(16t),8tsin(8t)-cos(8t)+8cos(16t)$:$)$