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DETAILS

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

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