Differentiation of ${\rm vers}\, v$

Let $y = {\rm vers}\ v$. By definition, this may be written

$$\ y = 1 - \cos v\ .$$

Differentiating, $\frac{dy}{dx} = \sin v \frac{dv}{dx}$. Therefore, $\frac{d}{dx}({\rm vers}\, v) = \sin v \frac{dv}{dx}$ (equation (XVII) in §5.1 above).