= Solution
With coincident origins and aligned axes, the <Lorentz transformation> is
$$
\boxed{x'=\gamma_v(x-vt),\qquad
t'=\gamma_v\left(t-\frac{vx}{c^2}\right),\qquad
\gamma_v=(1-v^2/c^2)^{-1/2}.}
$$
Its inverse replaces $v$ by $-v$. Along the worldline of the origin of $S''$, $dx'/dt'=u$. Differentiating the inverse <Lorentz transformation> gives
$$
dx=\gamma_v(dx'+vdt'),\qquad
dt=\gamma_v(dt'+v,dx'/c^2).
$$
Taking the ratio yields the <relativistic velocity-addition formula>
$$
\boxed{V=\frac{u+v}{1+uv/c^2}.}
$$
For $|u|,|v|\ll c$, the denominator tends to one, recovering \b[$V\simeq u+v$]. Simultaneously $\gamma_v\to1$, so $x'\simeq x-vt$ and $t'\simeq t$: these are the <Galilean transformation> and its Newtonian velocity rule.
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