My question pertains to the equation of hyperbolic motion in special relativity: $$x^{2} - c^{2}t^{2} = c^{4}/\alpha^{2}.$$ As far as I am aware, this equation is the key to calculating coordinate time for accelerating frame as they approach the speed of light. I cannot find a source for the derivation of this equation. I found some information at What is the relativistic calculation of travel time to Proxima Centauri? but it is pretty unclear as to the mathematical processes concerning rapidity.
The differentiation of this function:
$$\phi(u) = \phi(v) + \phi(u'),$$
into
$$\frac{\mathrm{d}}{\mathrm{d}t}\phi(u) = \frac{\mathrm{d}}{\mathrm{d}t'}\phi(u')\frac{\mathrm{d}t'}{\mathrm{d}t}$$
and the subsequent rewriting into
$$\frac{\mathrm{d}}{\mathrm{d}t}\phi(u) = \frac{1}{c}\gamma^{2}(u)\frac{\mathrm{d}u}{\mathrm{d}t},$$
and
$$\gamma^{3}(u')\frac{\mathrm{d}u'}{\mathrm{d}t'} = \gamma^{3}(u)\frac{\mathrm{d}u}{\mathrm{d}t}.$$
None of the process is particularly clear. Of course, the question above could be entirely wrong and there could be a much simpler derivation.