Answer
$\boldsymbol{r'}(t)=\langle2t,-2t\sin{t^2},2\sin{t}\cos{t}\rangle$.
Work Step by Step
$\boldsymbol{r}(t)=\langle t^2,\cos{t^2},\sin^2{t}\rangle$
In order to compute $\boldsymbol{r'}(t)$ we simply take the derivative of each component with respect to t of $\boldsymbol{r}(t)$.
$\boldsymbol{r'}(t)=\frac{d}{dt}\boldsymbol{r}(t)=\frac{d}{dt}\langle t^2,\cos{t^2},\sin^2{t}\rangle=\langle \frac{d}{dt}t^2,\frac{d}{dt}\cos{t^2},\frac{d}{dt}\sin^2{t}\rangle=\langle2t,-2t\sin{t^2},2\sin{t}\cos{t}\rangle$