Alternative formula for the affine connection in a new coordinate basis In Hobsons's General Relativity: An Introduction for Physicists, pg. 64,
he gave two different expressions for the affine connection $\Gamma'^a_{bc}$ in a transformed coordinate basis $x'^a$ (the original coodinate basis is $x^a$).
The two expressions are
$$\Gamma'^a_{bc}=\frac{\partial x'^a}{\partial x^d}\frac{\partial x^f}{\partial x'^b}\frac{\partial x^g}{\partial x'^c}\Gamma^d_{fg}+\frac{\partial x'^a}{\partial x^d}\frac{\partial ^2x^d}{\partial x'^c \partial x'^b},$$
$$\Gamma'^a_{bc}=\frac{\partial x'^a}{\partial x^d}\frac{\partial x^f}{\partial x'^b}\frac{\partial x^g}{\partial x'^c}\Gamma^d_{fg}-\frac{\partial x^d}{\partial x'^b}\frac{\partial x^f}{\partial x'^c}\frac{\partial^2 x'^a}{\partial x^d \partial x^f}.$$
Why are these two expressions the same?
Hobson said that the second expression is obtained by swapping derivative with respect to $x$ and $x'$ in the last term on the RHS of the first expression. I have a hard time seeing what he explicitly meant by that. The presence of a minus sign in the second expression confuses me.
 A: The last term in second expression can found from first by following manipulation:
$$\frac{\partial x'^a}{\partial x^d}\frac{\partial ^2x^d}{\partial x'^c \partial x'^b}$$
$$=\frac{\partial}{\partial x'^c}\bigg(\frac{\partial x'^a}{\partial x^d}\frac{\partial x^d}{\partial x'^b}\bigg)-\frac{\partial x^d}{\partial x'^b}\bigg[\frac{\partial}{\partial x'^c}\bigg]\bigg(\frac{\partial x'^a}{\partial x^d}\bigg)$$
$$=\frac{\partial}{\partial x'^c}\bigg(\frac{\partial x'^a}{\partial x'^b}\bigg)-\frac{\partial x^d}{\partial x'^b}\bigg[\frac{\partial x^f}{\partial x'^c}\frac{\partial}{\partial x^f}\bigg]\bigg(\frac{\partial x'^a}{\partial x^d}\bigg)$$
$$=\frac{\partial}{\partial x'^c}\Big(\delta^{'a}_{'b}\Big)-\frac{\partial x^d}{\partial x'^b}\frac{\partial x^f}{\partial x'^c}\frac{\partial^2 x'^a}{\partial x^d \partial x^f}$$
$$0-\frac{\partial x^d}{\partial x'^b}\frac{\partial x^f}{\partial x'^c}\frac{\partial^2 x'^a}{\partial x^d \partial x^f}$$
There is a subtle point I have left in the above answer; try to figure it out. It's explained in the last.
Now for why he went for second expression. If you have probably tried to find how $\partial_{\alpha}V^{\beta}$ transforms during a coordinate change you'll find there is an additional term that pops out, it's same as above term with opposite sign. This is the reason why define $\partial_{\alpha}V^{\beta}+\Gamma^\beta_{\alpha\gamma}V^\gamma$ as some kind of derivative which transforms as a tensor during coordinate change. The nontensorial part cancels each other out. So the reason is just a coherence of expression, nothing more. Otherwise, both expressions are inherently fundamental.

! Missing point $$\bigg[\frac{\partial}{\partial x'^c}\bigg]\bigg(\frac{\partial x'^a}{\partial x^d}\bigg)$$
$$\neq\bigg[\frac{\partial}{\partial x^d}\bigg]\bigg(\frac{\partial x'^a}{\partial x'^c}\bigg)$$
$$\neq \frac{\partial}{\partial x^d}\delta^{'a}_{'c}$$
$$\neq 0$$ The issue lies in the second line and our sloppiness in notation of partial derivative
$$\bigg[\frac{\partial}{\partial x'^c(o)}\bigg]_{\mathrm{n\hspace{2pt}is\hspace{2pt}kept\hspace{2pt}constant}}\bigg(\frac{\partial x'^a(o)}{\partial x^d(n)}\bigg)_{\mathrm{o\hspace{2pt}is\hspace{2pt}kept\hspace{2pt}constant}}$$
where n stands for new coordinate, o for old coordinates.

