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