# Reparametrization by Arclength Definition

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?

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 $$s'(t(s)) t'(s)= 1, \quad \text{ or } \quad t'(s) = 1/s'(t(s)).$$