Derive the Lemma of Christoffel symbol, $$\frac{\partial^2x^\rho}{\partial\bar x^\mu\partial\bar x^\nu}=\Gamma_{\mu\nu}^\gamma\frac{\partial x^\rho}{\partial\bar x^\gamma}-\frac{\partial x^\alpha}{\partial\bar x^\mu}\frac{\partial x^\beta}{\partial\bar x^\nu}\Gamma_{\alpha\beta}^\rho$$
We know that, $$\bar g_{ij}=\frac{\partial x^\alpha}{\partial \bar x^i}\frac{\partial x^\beta}{\partial\bar x^j}g_{\alpha\beta}\tag 1$$ Now, the process my book followed, differentiate $(1)$ with respect to different co-ordinate $\bar x^i$, interchange several variable and use the definition of Christoffel symbol of first kind and multiplying conjugate metric $\bar g^{rk}$. The complete computation was so tedious and it took 5 pages. Now, I can't just memorize that and copy in my exam sheet. Is there any other approach where things were flowing a sequence or I can easily memorize or understand those steps intuitively.
Any solution or external redirect will be appreciated. Thanks in advance.