Answer
$\mathrm{u}^{\prime}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}(t)]-\mathrm{v}^{\prime}(t)\cdot[\mathrm{u}(t)\times \mathrm{w}(t)]+\mathrm{w}^{\prime}(t)\cdot[\mathrm{u}(t)\times \mathrm{v}(t)]$
Work Step by Step
Apply Theorem 3. First, formula 4
$\displaystyle \frac{d}{dt}(\mathrm{u}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}(t)])=\mathrm{u}^{\prime}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}(t)]+\mathrm{u}(t)\cdot\frac{d}{dt}[\mathrm{v}(t)\times \mathrm{w} (t)]$
... then, formula 5,
$=\mathrm{u}^{\prime}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}(t)]+\mathrm{u}(t)\cdot[\mathrm{v}^{\prime}(t)\times \mathrm{w}(t)+\mathrm{v}(t)\times \mathrm{w}^{\prime}(t)]$
... now, the distributive property of the cross product (sec.12-4, Th.11)
$=\mathrm{u}^{\prime}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}(t)]+\mathrm{u}(t)\cdot[\mathrm{v}^{\prime}(t)\times \mathrm{w}(t)]+\mathrm{u}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}^{\prime}(t)]$
... and property 1 in Th.11, sec.12-4
$=\mathrm{u}^{\prime}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}(t)]-\mathrm{u}(t)\cdot[\mathrm{w}(t)\times \mathrm{v}^{\prime}(t)]+\mathrm{u}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}^{\prime}(t)]$
... and property 5 in Th.11, sec.12-4
$=\mathrm{u}^{\prime}(t)\cdot[\mathrm{v}(t)\times \mathrm{w}(t)]-\mathrm{v}^{\prime}(t)\cdot[\mathrm{u}(t)\times \mathrm{w}(t)]+\mathrm{w}^{\prime}(t)\cdot[\mathrm{u}(t)\times \mathrm{v}(t)]$
