Jacobian of a diffeomorphism $\phi:\mathbb{S}^2\to\mathbb{S}^2$

Question

I'm trying to understant exercise $(4)$, chapter $5$, from Montiel-Ros's Curves and Surfaces to which there is a hint at the end of the chapter:

Hint:

I've understood everything from the hint, except for the last two lines. More precisely, this equality: $$|\text{Jac}(\phi)|(p)=\frac{|\det(Ae_1\,Ae_2\,Ap)|}{|Ap|^3}$$

How did he conclude this from the expression of $(d\phi)_p(v)$?