In my textbook, reparametrization by arclength is described as follows:
Take a regular vector function $\alpha (t)$ whose arclength is $s(t)$. $s(t)$ is an increasing differentiable function, so it has a differentiable inverse function $t=t(s)$. So consider the parametrization $\beta (s) = \alpha (t(s))$. By the chain rule, $\beta'(s) = \alpha'(t(s))t'(s) = \alpha'(t(s))/s'(t(s)) = \alpha'(t(s))/\left\lVert \alpha '(t(s)) \right\rVert$.
I don't see where the 2nd equality comes from. It would imply that $t'(s) = 1/s'(t(s))$, and I just don't see where that comes from. This is the first section in the book, so it's not from an earlier part. So where does this equality come from?