Deriving calculation formulas for torsion and curvature
It shouldn't be "a mindless differentiation bash" if you write down the equation $\alpha' = \upsilon T$, where $\upsilon=\|\alpha'\|$, use the Frenet equations and chain rule to differentiate twice. It helps to economize a bit by remembering that $w\times w=0$ for any vector $w$. In particular, $\alpha'\times\alpha''$ will point in the $T\times N= B$ direction, so you'll need only the $B$ component of $\alpha'''$. (P.S.@Jesse Madnick, I don't think the identity for $\|v\times w\|$ is needed at all.)