# Divergence in orthogonal curvilinear coordinate system question

I'm trying to understand how the divergence formula in curvilinear coordinates is derived, but unfortunately my textbook doesn't go into much detail. Here is what they show:

I think I understand how they get the LHS of (8.9) but I don't know how they get to the RHS. Can someone please show me how the LHS equals the RHS?

$$f(x+dx)=f(x)+f'(x)dx +[\text{terms of second and greater degree}]$$
so is, $f(x+dx)\approx f(x)+f'(x)dx$ or $f(x+dx)-f(x)\approx f'(x)dx$
$A_1h_2h_3|_{x^1x^2x^3}$ is some function of the three variables, say $g(x^1,x^2,x^3)$, then, we, following more strictly than the author the notation he chose, can write: $\left.\dfrac{\partial(A_1h_2h_3)}{\partial x^1}\right|_{(x^1,x^2,x^3)}=\dfrac{\partial g(x^1,x^2,x^3)}{\partial x^1}$, so is, the derivative wrt $x^1$ evaluated at the point $(x^1,x^2,x^3)$
You surely can complete the "translation" (remember that $x^2$ and $x^3$ act as simple constants here).