Answer
$\mathrm{r}^{\prime}(t)=\mathrm{b}+(2t)\mathrm{c}$
Work Step by Step
Apply Th.3.1 and Th.3.3:
$\displaystyle \frac{d}{dt}[\mathrm{u}(t)+\mathrm{v}(t)]=\mathrm{u}^{\prime}(t)+\mathrm{v}^{\prime}(t),$
$\displaystyle \frac{d}{dt}[f(t)\cdot \mathrm{u}(t)]=f^{\prime}(t)\cdot \mathrm{u}(t)+f(t)\cdot \mathrm{u}^{\prime}(t)$
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$\displaystyle \frac{d}{dt}[a]=0\qquad$ (const)
$\displaystyle \frac{d}{dt}[t\mathrm{b}]=(1)\mathrm{b}+t(0)=\mathrm{b}$
$\displaystyle \frac{d}{dt}[t^{2}\mathrm{c}]=2t\mathrm{c}+t^{2}(0)=2t\mathrm{c}$
$\mathrm{r}^{\prime}(t)=0+\mathrm{b}+2t\mathrm{c}$
$\mathrm{r}^{\prime}(t)=\mathrm{b}+2t\mathrm{c}$
