Transformation of Christoffel Symbol

I’m studying General Relativity by Hobson and got stuck in understanding the derivation of the transformation law of Christoffel symbols (p. 64). Let $x^a$ and $x’^a$ be two coordinate systems on a manifold. In this book, the Christoffel symbol (or the affine connection) is defined by $$frac{partial vec{e}_a}{partial x^c} = Gamma^b_{,ac} vec{e}_b tag{3.12}$$ where $vec{e}_a$ denote the basis vectors corresponding to the $x_a$ coordinates. After some calculations, the book reaches the transformation law below which can be found in many other sources, e.g. equation (10) here:
$$Gamma’^a_{,,bc} = vec{e}’^a cdot frac{partial vec{e}’_b}{partial x’^c} = cdots = frac{partial x’^a}{partial x^d}frac{partial x^f}{partial x’^b}frac{partial x^g}{partial x’^c}Gamma^d_{,fg} + frac{partial x’^a}{partial x^d}frac{partial^2 x^d}{partial x’^c partial x’^b} ,. tag{3.16}$$

This is all very well, but I have a trouble with the next step. The book says that by swapping derivatives with respect to $x$ and $x’$ in the last term on the right-hand side of (3.16), we can arrive at an alternative but equivalent expression:
$$ Gamma’^a_{,,bc} = frac{partial x’^a}{partial x^d}frac{partial x^f}{partial x’^b}frac{partial x^g}{partial x’^c}Gamma^d_{,fg} – frac{partial x^d}{partial x’^b}frac{partial x^f}{partial x’^c}frac{partial^2 x’^a}{partial x^d partial x^f} ,. tag{3.17}$$

My question is:

1. I cannot figure out how the last term in (3.16) can be converted to that in (3.17). Where does the minus sign come from?
2. I am skeptical about the last term in (3.17) because I reached the same conclusion with this post that this term vanishes. The author actually uses equation (3.17) in p.68 of the textbook, so I’m really confused.

I appreciate any help.