I am reading Do Carmo's Differential Geometry of Curves and Surfaces, and I am stuck understanding why $|\beta'(s)|=|\alpha'(t)\cdot (dt/ds)| = 1$ is true.
For context, here is the paragraph where it is reparameterizing a curve $\alpha(t)$ by its arc length. I can follow all the argument, but I am stuck on the above result.
$\left|\frac{d}{ds} \beta(s)\right| = \left|\frac{d}{ds} \alpha(t(s))\right|=|\alpha'(t) \cdot t'(s)|$.
How does this equal $1$?
He writes that $|\alpha'(t)|=s'(t)$, and since $t(s)$ and $s(t)$ are inverses, their derivatives are related via $$t'(s)=\frac{1}{s'(t)}.$$ Plugging this in gives $1.$ See https://en.wikipedia.org/wiki/Inverse_functions_and_differentiation.