Suppose that, as seen from $O$, $O'$ is moving to the right with speed $v$ and that an object is moving to the right with speed $u$. According to Galileo we would simply add velocities to determine the velocity of the object as seen from $O'$: \begin{equation} u=u'+v \end{equation} This equation can be derived by differentiating the Galilean transformation \begin{equation} x=x'+vt \end{equation} thus obtaining \begin{equation} \frac{dx}{dt} = \frac{dx'}{dt}+v \end{equation}

To derive the relativistic formula for the addition of velocities, we proceed similarly. However, since both $x$ and $t$ transform, it is useful to use the differential form of the Lorentz transformations, namely \begin{eqnarray} dx &=& d\,\Big(\,\gamma\,(x'+vt') \Big) = \gamma \,(dx'+v\,dt') \\ dt &=& d\,\Big(\,\gamma\,(t'+\vcsq x') \Big) = \gamma \left(dt'+\vcsq dx'\right) \end{eqnarray} Dividing these expressions leads to \begin{equation} \frac{dx}{dt} = \frac{\frac{dx'}{dt'}+v}{1+\vcsq\frac{dx'}{dt'}} \end{equation} or equivalently \begin{equation} u=\frac{u'+v}{1+\overcc{u'v}} \label{padd} \end{equation}

Equation ($\ref{padd}$) is known as the Einstein addition formula, and shows that the addition of relativistic velocities does not obey our intuitive notion of how velocities should add. It is worth considering special cases of this formula, such as when both $u'$ and $v$ are small compared to $c$, or when one of them is equal to $c$. We will provide a geometric explanation for the apparently peculiar properties below.