Solution
= Solution
To first order in $V/c$, $\gamma=1+O(V^2/c^2)$ and
$$
\frac1{1-Vu_x/c^2}
=1+\frac{Vu_x}{c^2}+O(V^2/c^2).
$$
Thus
$$
u'_x=u_x-V+\frac{Vu_x^2}{c^2}+O(V^2/c^2),
\qquad
u'_y=u_y+\frac{Vu_xu_y}{c^2}+O(V^2/c^2).
$$
Since $\mathbf V=(V,0,0)$ and $\mathbf V\cdot\mathbf u=Vu_x$, these components combine into
$$
\boxed{
\mathbf u'=\mathbf u-\mathbf V
+\frac{\mathbf V\cdot\mathbf u}{c^2}\mathbf u
+O(V^2/c^2)}.
$$