I was answering a question which required calculating a double covariant derivative of a vector ($ D_\mu D_\nu V^\rho$).
But then I got stuck trying to apply a normal partial derivative to a Christoffel symbol defined as $$\Gamma^\rho_{\mu\nu} = \frac{1}{2} g^{\rho k} \bigg[\frac{\partial g_{k\mu}}{\partial q^\nu} + \frac{\partial g_{k\nu}}{\partial q^\mu} - \frac{\partial g_{\nu\mu}}{\partial q^k} \bigg],$$
but when I try to differentiate it gives another derivative of Christoffel symbol which inserts me in an endless loop with apparently no way out.
Does there exist a way to compute derivatives of Christoffel symbols, or is it mathematically wrong to do that for some reason I am unaware of?
It is impossible to derive the derivative of Christoffel symbol only in terms of metric and Christoffel symbols themself.
If it was possible, the stationary surface, determined by the Einstain equation for vacuum, could be parametrised by only metric tensor and Christoffel symbols. But we know, that in field theory in general case there are infinite amount of independent variables are exist on this surface (unlike the mechanic case, in which you can determine any derivative of coordinates and momenta in terms of coordinates and momenta themself).
However, it can be calculated using Leibniz identity.