Answer
$\mathrm{r}^{\prime}(t)=(\mathrm{a}\times \mathrm{b})+2t(\mathrm{a}\times \mathrm{c})$.
Work Step by Step
Use Th.11 from sec 12-4 (properties of the cross product)
$\mathrm{r}(t)=t\mathrm{a}\times(\mathrm{b}+t\mathrm{c}),\qquad $...Th.11.4
$\mathrm{r}(t)=(t\mathrm{a}\times \mathrm{b})+(t\mathrm{a}\times t\mathrm{c}),\qquad $...Th.11.2
$\mathrm{r}(t)=t(\mathrm{a}\times \mathrm{b})+t^{2}(\mathrm{a}\times \mathrm{c})$
Now, apply Th.3.1 and Th.3.3:
$\displaystyle \frac{d}{dt}[\mathrm{u}(t)+\mathrm{v}(t)]=\mathrm{u}^{\prime}(t)+\mathrm{v}^{\prime}(t),$
$\displaystyle \frac{d}{dt}[f(t)\cdot \mathrm{u}(t)]=f^{\prime}(t)\cdot \mathrm{u}(t)+f(t)\cdot \mathrm{u}^{\prime}(t)$
$\displaystyle \frac{d}{dt}[t(\mathrm{a}\times \mathrm{b})]= (1)(\mathrm{a}\times \mathrm{b})+t(0)=\mathrm{a}\times \mathrm{b}$
$\displaystyle \frac{d}{dt}[t^{2}(\mathrm{a}\times \mathrm{c})]= (2t)(\mathrm{a}\times \mathrm{c})+t^{2}(0)=2t(\mathrm{a}\times \mathrm{c})$
$\mathrm{r}^{\prime}(t)=\mathrm{a}\times \mathrm{b}+2t(\mathrm{a}\times \mathrm{c})$.
