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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?

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It follows from the chain rule. Since $t=t(s)$ and $s=s(t)$ are inverse functions, $$ s(t(s))= s. $$ Deriving with respect to $s$ gives $$ s'(t(s)) t'(s)= 1, \quad \text{ or } \quad t'(s) = 1/s'(t(s)). $$

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