Differentiation of the connections

[JimWhoKnew]

I'm reading this paper where the authors implicitly differentiate connections along a worldline$$\frac{D~\Gamma^\mu{}_{\nu\rho}}{d\tau} \quad. \tag{1}$$So far, it seems that I can recover their results if I calculate "$\Gamma^\mu{}_{\nu\rho;\sigma}$" as if the connections were components of an ordinary (1,2) tensor. I'm not confident it's OK to do so, and couldn't find (so far) any example in literature.

What is the correct way to evaluate (1) when the metric and connections on the worldline are known as functions of the proper time $\tau$, and why?

 

[PeterDonis]

What is the correct way to evaluate (1) when the metric and connections on the worldline are known as functions of the proper time $\tau$, and why?

Um, just differentiate the known functions of $\tau$ with respect to $\tau$?

 

[JimWhoKnew]

Um, just differentiate the known functions of $\tau$ with respect to $\tau$?

First, by doing so I fail to recover the results of the mentioned paper.
Second, there are basis vectors and covectors involved. Why are we allowed to ignore their changes?
As I understand your reply, you suggest$$\frac{D~\Gamma^\mu{}_{\nu\rho}}{d\tau} = \frac{\partial~\Gamma^\mu{}_{\nu\rho}}{\partial\tau}\quad.$$right?

Edit:
Section 13.6 in MTW discusses "detector's proper frame". The linked paper extends the results to second order in $x^\mu$ (proper deviations from the relevant point on the worldline).