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TeX (original user input):

\begin{align}
  & \Gamma '_{\nu \lambda }^{\mu }=\frac{\partial x{{'}^{\mu }}}{\partial {{\xi }^{\tau }}}\frac{{{\partial }^{2}}{{\xi }^{\tau }}}{\partial x{{'}^{\nu }}\partial x{{'}^{\lambda }}} \\ 
 & =\frac{\partial x{{'}^{\mu }}}{\partial {{\xi }^{\sigma }}}\frac{\partial {{x}^{\sigma }}}{\partial {{\xi }^{\tau }}}\frac{\partial }{\partial x{{'}^{\nu }}}\left( \frac{\partial {{\xi }^{\tau }}}{\partial {{x}^{\alpha }}}\frac{\partial {{x}^{\alpha }}}{\partial x{{'}^{\lambda }}} \right) \\ 
 & =\frac{\partial x{{'}^{\mu }}}{\partial {{\xi }^{\sigma }}}\frac{\partial {{x}^{\sigma }}}{\partial {{\xi }^{\tau }}}\left( \frac{{{\partial }^{2}}{{\xi }^{\tau }}}{\partial {{x}^{\alpha }}\partial {{x}^{\rho }}}\frac{\partial {{x}^{\rho }}}{\partial {{x}^{\alpha }}}\frac{\partial {{x}^{\alpha }}}{\partial x{{'}^{\lambda }}}+\frac{{{\partial }^{2}}{{x}^{\alpha }}}{\partial x{{'}^{\nu }}\partial x{{'}^{\lambda }}}\frac{\partial {{\xi }^{\tau }}}{\partial {{x}^{\alpha }}} \right) \\ 
 & =\frac{\partial x{{'}^{\mu }}}{\partial {{\xi }^{\sigma }}}\frac{\partial {{x}^{\sigma }}}{\partial {{\xi }^{\tau }}}\frac{\partial {{x}^{\alpha }}}{\partial x{{'}^{\lambda }}}\Gamma _{\alpha \rho }^{\sigma }+\frac{\partial x{{'}^{\mu }}}{\partial {{x}^{\sigma }}}\frac{{{\partial }^{2}}{{x}^{\sigma }}}{\partial x{{'}^{\nu }}\partial x{{'}^{\lambda }}}  
\end{align}

TeX (checked):

{\begin{aligned}&\Gamma '_{\nu \lambda }^{\mu }={\frac {\partial x{{'}^{\mu }}}{\partial {{\xi }^{\tau }}}}{\frac {{{\partial }^{2}}{{\xi }^{\tau }}}{\partial x{{'}^{\nu }}\partial x{{'}^{\lambda }}}}\\&={\frac {\partial x{{'}^{\mu }}}{\partial {{\xi }^{\sigma }}}}{\frac {\partial {{x}^{\sigma }}}{\partial {{\xi }^{\tau }}}}{\frac {\partial }{\partial x{{'}^{\nu }}}}\left({\frac {\partial {{\xi }^{\tau }}}{\partial {{x}^{\alpha }}}}{\frac {\partial {{x}^{\alpha }}}{\partial x{{'}^{\lambda }}}}\right)\\&={\frac {\partial x{{'}^{\mu }}}{\partial {{\xi }^{\sigma }}}}{\frac {\partial {{x}^{\sigma }}}{\partial {{\xi }^{\tau }}}}\left({\frac {{{\partial }^{2}}{{\xi }^{\tau }}}{\partial {{x}^{\alpha }}\partial {{x}^{\rho }}}}{\frac {\partial {{x}^{\rho }}}{\partial {{x}^{\alpha }}}}{\frac {\partial {{x}^{\alpha }}}{\partial x{{'}^{\lambda }}}}+{\frac {{{\partial }^{2}}{{x}^{\alpha }}}{\partial x{{'}^{\nu }}\partial x{{'}^{\lambda }}}}{\frac {\partial {{\xi }^{\tau }}}{\partial {{x}^{\alpha }}}}\right)\\&={\frac {\partial x{{'}^{\mu }}}{\partial {{\xi }^{\sigma }}}}{\frac {\partial {{x}^{\sigma }}}{\partial {{\xi }^{\tau }}}}{\frac {\partial {{x}^{\alpha }}}{\partial x{{'}^{\lambda }}}}\Gamma _{\alpha \rho }^{\sigma }+{\frac {\partial x{{'}^{\mu }}}{\partial {{x}^{\sigma }}}}{\frac {{{\partial }^{2}}{{x}^{\sigma }}}{\partial x{{'}^{\nu }}\partial x{{'}^{\lambda }}}}\end{aligned}}

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\begin{aligned} Γ'_{ν λ}^{μ} = \frac{\partial x'^{μ}}{\partial ξ^{τ}} \frac{\partial^{2} ξ^{τ}}{\partial x'^{ν} \partial x'^{λ}} \\ = \frac{\partial x'^{μ}}{\partial ξ^{σ}} \frac{\partial x^{σ}}{\partial ξ^{τ}} \frac{\partial}{\partial x'^{ν}} (\frac{\partial ξ^{τ}}{\partial x^{α}} \frac{\partial x^{α}}{\partial x'^{λ}}) \\ = \frac{\partial x'^{μ}}{\partial ξ^{σ}} \frac{\partial x^{σ}}{\partial ξ^{τ}} (\frac{\partial^{2} ξ^{τ}}{\partial x^{α} \partial x^{ρ}} \frac{\partial x^{ρ}}{\partial x^{α}} \frac{\partial x^{α}}{\partial x'^{λ}} + \frac{\partial^{2} x^{α}}{\partial x'^{ν} \partial x'^{λ}} \frac{\partial ξ^{τ}}{\partial x^{α}}) \\ = \frac{\partial x'^{μ}}{\partial ξ^{σ}} \frac{\partial x^{σ}}{\partial ξ^{τ}} \frac{\partial x^{α}}{\partial x'^{λ}} Γ_{α ρ}^{σ} + \frac{\partial x'^{μ}}{\partial x^{σ}} \frac{\partial^{2} x^{σ}}{\partial x'^{ν} \partial x'^{λ}} \end{aligned}

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Identifiers

$Γ$

$ν$

$λ$

$μ$

$x$

$μ$

$ξ$

$τ$

$ξ$

$τ$

$x$

$ν$

$x$

$λ$

$x$

$μ$

$ξ$

$σ$

$x$

$σ$

$ξ$

$τ$

$x$

$ν$

$ξ$

$τ$

$x$

$α$

$x$

$α$

$x$

$λ$

$x$

$μ$

$ξ$

$σ$

$x$

$σ$

$ξ$

$τ$

$ξ$

$τ$

$x$

$α$

$x$

$ρ$

$x$

$ρ$

$x$

$α$

$x$

$α$

$x$

$λ$

$x$

$α$

$x$

$ν$

$x$

$λ$

$ξ$

$τ$

$x$

$α$

$x$

$μ$

$ξ$

$σ$

$x$

$σ$

$ξ$

$τ$

$x$

$α$

$x$

$λ$

$Γ$

$α$

$ρ$

$σ$

$x$

$μ$

$x$

$σ$

$x$

$σ$

$x$

$ν$

$x$

$λ$

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