Answer
$\mathrm{r}^{\prime}(t)= (a\cos 3t -\mathit{3}at \sin 3t)\mathrm{i}+(3b\sin^{2}t \cos t)\mathrm{j} -(3c\cos^{2}t\sin t)\mathrm{k}$
Work Step by Step
Parametric equations: $\mathrm{r}(t):\quad\left\{\begin{array}{ll}
x=at\cos 3t & \\
y=b\sin^{3}t & \\
y=c\cos^{3}t &
\end{array}\right.$
Differentiate $(\displaystyle \frac{d}{dt})$ each component function.
x: product and chain rules
y, z, : chain rule
$\mathrm{r}^{\prime}(t):\quad\left\{\begin{array}{ll}
x=at(-\sin 3t\cdot 3) +(a)\cos 3t & =a\cos 3t -\mathit{3}at \sin 3t\\ \\
y=b\cdot 3\sin^{2}t \cdot\cos t\\ & \\
z=c\cdot 3\cos^{2}t(-\sin t) &
\end{array}\right. $
$\mathrm{r}^{\prime}(t)= (a\cos 3t -\mathit{3}at \sin 3t)\mathrm{i}+(3b\sin^{2}t \cos t)\mathrm{j} -(3c\cos^{2}t\sin t)\mathrm{k}$
