Proof that the unit tangent vector has length $1$?

Question

I want to prove that $|dr/dS| = 1$. I know that this can be proven using the chain rule, I'm just not sure how. Any help would be appreciated.

Answer 1

Starting: $S(x) = \int_{a}^x ||r'(t)||dt$. Differentiate both sides w.r.t $x$:

$S'(x) = ||r'(x)||$. Thus: $1 = \left|\dfrac{r'(x)}{S'(x)}\right| = \left|\dfrac{\dfrac{dr}{dx}}{\dfrac{dS}{dx}}\right| = \left|\dfrac{dr}{dS}\right|$