If $u(t)=($:$sin(7t),cos(7t),t$:$)$ and $v(t)=($:$t,cos(7t),sin(7t)$:$),$ use the theorem given below to find $\frac{d}{dt}[u(t)v(t)].$

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

$(7co{s}^2(7t)+7tsin(7t))i-(7sin(7t)cos(7t)-t)j+(-7si{n}^2((7t)-cos(7t))k)$