Answer
The derivative at $t=3$:
$\frac{d}{{dt}}\left( {{{\bf{r}}_1}\left( t \right) \times {{\bf{r}}_2}\left( t \right)} \right){|_{t = 3}} = \left( {3, - 3,2} \right)$
Work Step by Step
$\frac{d}{{dt}}\left( {{{\bf{r}}_1}\left( t \right) \times {{\bf{r}}_2}\left( t \right)} \right) = \left( {\frac{d}{{dt}}{{\bf{r}}_1}\left( t \right)} \right) \times {{\bf{r}}_2}\left( t \right) + {{\bf{r}}_1}\left( t \right) \times \left( {\frac{d}{{dt}}{{\bf{r}}_2}\left( t \right)} \right)$
$ = {{\bf{r}}_1}'\left( t \right) \times {{\bf{r}}_2}\left( t \right) + {{\bf{r}}_1}\left( t \right) \times {{\bf{r}}_2}'\left( t \right)$
The derivative at $t=3$:
$\frac{d}{{dt}}\left( {{{\bf{r}}_1}\left( t \right) \times {{\bf{r}}_2}\left( t \right)} \right){|_{t = 3}} = {{\bf{r}}_1}'\left( 3 \right) \times {{\bf{r}}_2}\left( 3 \right) + {{\bf{r}}_1}\left( 3 \right) \times {{\bf{r}}_2}'\left( 3 \right)$
$ = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
0&0&1\\
1&1&0
\end{array}} \right| + \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
1&1&0\\
0&2&4
\end{array}} \right|$
$ = - {\bf{i}} + {\bf{j}} + 4{\bf{i}} - 4{\bf{j}} + 2{\bf{k}}$
$ = 3{\bf{i}} - 3{\bf{j}} + 2{\bf{k}}$
$\frac{d}{{dt}}\left( {{{\bf{r}}_1}\left( t \right) \times {{\bf{r}}_2}\left( t \right)} \right){|_{t = 3}} = \left( {3, - 3,2} \right)$
