If $u(t)=($:$sin(5t),cos(3t),t$:$)$ and $v(t)=($:$t,cos(3t),sin(5t)$:$),$ use the theorem given below to find $\frac{d}{dt}[u(t)*v(t)].$

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