Answer
As a result $\boldsymbol{r'}(t)=\boldsymbol{b}+2t\boldsymbol{c}$.
Work Step by Step
$\boldsymbol{r}(t)=\boldsymbol{a}+t\boldsymbol{b}+t^2\boldsymbol{c}$
In order to compute $\boldsymbol{r'}(t)$ we simply take the derivative of each component with respect to t of $\boldsymbol{r}(t)$.
$\boldsymbol{r'}(t)=\frac{d}{dt}\boldsymbol{a}+\frac{d}{dt}t\boldsymbol{b}+t^2\boldsymbol{c}=0\boldsymbol{a}+\boldsymbol{b}+2t\boldsymbol{c}$