Answer
$r'(2)=12i-29j+14k$
or
$r'(2)=\lt 12,-29,14 \gt$
Work Step by Step
Given: $r(t)=u(t) \times v(t), u(2)=\lt 1,2,-1 \gt,u'(2)=\lt 3,0,4 \gt$ and $v(t)=\lt t, t^2,t^3 \gt$
Our aim is to calculate $r'(2)$.
In order to find this, we will use the cross product rule.
$r'(t)=u'(t) \times v(t)+u(t) \times v'(t)$
$v(t)=\lt t, t^2,t^3 \gt \implies v'(t)=\lt 1, 2t,3t^2 \gt$
and $v'(2)=\lt 1,4,12 \gt$
Thus, $r'(2)=\lt 3,0,4 \gt \times \lt 2,4,8 \gt+\lt 1,2,-1 \gt \times \lt 1,4,12\gt$
$=\begin{vmatrix}1&j&k \\3&0&4\\2&4&8 \end{vmatrix}+\begin{vmatrix}1&j&k \\1&2&-1\\1&4&12 \end{vmatrix}$
$=(-16i-16j+12k)+(28i-13j+2k)$
$=(-16+28)i+(-16-13)j+(12+2)k$
Hence, $r'(2)=12i-29j+14k$
or
$r'(2)=\lt 12,-29,14 \gt$
