I've scoured this site, the web, and some books for derivations of the formula for proper acceleration as a function of coordinate acceleration ($\alpha = \gamma^3 a$) without four vectors or hyperbolic trig functions. I've found a number of them but they all skip so many steps that they're impossible for me to fully follow. The most complete attempt I've seen is this one (see the "3-vector approach" section), and I can follow all of it except for part of one step—where it implies that
$$ \frac{d}{dt}\frac{u_x - v}{1-\frac{vu_x}{c^2}} = \frac{1}{\gamma^2\left(1-\frac{vu_x}{c^2}\right)^2}\frac{d^2x}{dt^2} $$
I can't see how to get from the left side to the right side, so can you either break it down for me or provide another derivation altogether? I realize that proper acceleration can also be derived using four vectors or rapidity but neither of those are yet intuitive enough for me to give me a complete understanding of proper acceleration and how to use it or to fully convince me that the formula $\alpha = \gamma^3 a$ is correct. I'm fine with assuming that all acceleration is in the $x$ direction and parallel to the velocity.
The next best thing would be an explanation of why the formula should make sense intuitively. The next would be a four vector approach that's simple and fully spelled out. For example, this one was a good start but the norm of the four acceleration it ends with doesn't seem to give the expected result (not that it's wrong; I'm probably just missing something).