Jacobian identity used in proof of change of variables

Question

This is from "Calculus on Manifolds", proof of Change of Variables theorem. I don't understand why these two red circled expressions are equal. $$|det(h\circ g)'|=|det(h'\circ g)||det(g')|$$ by Chain Rule, but I see no way this can be further simplified to $(|det(h')|\circ g)\ \cdot |det(g')|$

Answer 1

Just write it down: $$\big(|\det h'|\circ g\big)(x) = |\det h'(g(x))| = |\det(h'\circ g)(x)|.$$ The $|\det g'|$ comes from the chain rule and multiplicativity of determinant: $$|\det (h\circ g)'(x)| = |\det \big(h'(g(x))g'(x)\big)| = |\det h'(g(x))||\det g'(x)| = |\det(h'\circ g)(x)||\det g'(x)|,$$ as needed.

Answer 2

I'll add my answer in case someone's faced with the same trouble. It is fust about the order of functional composition. In both cases we first apply $g(x)$, then $h'(x)$, then $det(x)$ and finally $|x|$