Can someone please check my work below to confirm whether or not I got the correct answer? This is question 13.2.16 in the 7th edition of Stewart Calculus.
Find the derivative of the vector function: $\vec{r}(t)=t\vec{a}\times(\vec{b}+t\vec{c})$
Use: $\frac{d}{dt}[\vec{u}(t)\times\vec{v}(t)]=\vec{u'}(t)\times\vec{v}(t)+\vec{u}(t)\times\vec{v'}(t)$
$\vec{u}(t)=t\vec{a}$
$\vec{v}(t)=(\vec{b}+t\vec{c})$
$\vec{u'}(t)=\vec{a}$
$\vec{v'}(t)=\vec{c}$
Thus, $\vec{r'}(t)=\vec{a}\times(\vec{b}+t\vec{c})+t\vec{a}\times\vec{c}$