Answer
$\mathrm{r}^{\prime}(t)=\langle t \cos t +\sin t$ ,$\quad 2t, \quad \cos 2t -2t \sin 2t \rangle$
Work Step by Step
Parametric equations: $\mathrm{r}(t):\quad\left\{\begin{array}{ll}
x=t\sin t & \\
y=t^{2} & \\
y=t\cos 2t &
\end{array}\right.$
Differentiate $(\displaystyle \frac{d}{dt})$ each component function
$\mathrm{r}^{\prime}(t):\quad\left\{\begin{array}{lll}
x=(1)\sin t+t(\cos t) & ... & \text{.. product rule} \\
y=2t & & \\
z=(1)\cos 2t+t(-2\sin 2t) & & \text{.. product and chain rules}
\end{array}\right.$
$\mathrm{r}^{\prime}(t):\quad\left\{\begin{array}{l}
x=t \cos t +\sin t\\
y=2t\\
z=\cos 2t -2t \sin 2t
\end{array}\right.$
$\mathrm{r}^{\prime}(t)=\langle t \cos t +\sin t$ ,$\quad 2t, \quad \cos 2t -2t \sin 2t \rangle$
