Answer
$\frac{d}{{ds}}{\bf{v}}\left( {g\left( s \right)} \right){|_{s = 4}} = - 54{\bf{i}} - 18{\bf{j}} + 6{\bf{k}}$
Work Step by Step
We have ${\bf{v}}\left( s \right) = {s^2}{\bf{i}} + 2s{\bf{j}} + 9{s^{ - 2}}{\bf{k}}$. So,
${\bf{v}}'\left( s \right) = 2s{\bf{i}} + 2{\bf{j}} - 18{s^{ - 3}}{\bf{k}}$
Using the Chain Rule on $\frac{d}{{ds}}{\bf{v}}\left( {g\left( s \right)} \right)$ gives
$\frac{d}{{ds}}{\bf{v}}\left( {g\left( s \right)} \right) = {\bf{v}}'\left( {g\left( s \right)} \right)g'\left( s \right)$
Evaluate $\frac{d}{{ds}}{\bf{v}}\left( {g\left( s \right)} \right)$ at $s=4$.
$\frac{d}{{ds}}{\bf{v}}\left( {g\left( s \right)} \right){|_{s = 4}} = {\bf{v}}'\left( {g\left( 4 \right)} \right)g'\left( 4 \right)$
Since $g\left( 4 \right) = 3$ and $g'\left( 4 \right) = - 9$, so
$\frac{d}{{ds}}{\bf{v}}\left( {g\left( s \right)} \right){|_{s = 4}} = - 9{\bf{v}}'\left( 3 \right) = - 9\left( {6{\bf{i}} + 2{\bf{j}} - \frac{{18}}{{27}}{\bf{k}}} \right)$
$\frac{d}{{ds}}{\bf{v}}\left( {g\left( s \right)} \right){|_{s = 4}} = - 54{\bf{i}} - 18{\bf{j}} + 6{\bf{k}}$
