Problems with the Riemann tensor in general relativity

[Ineedhelpimbadatphys]

Homework Statement

In the picture.

Relevant Equations

Also in pictures.

After Taylor expansion and using equations (2), I have no problem getting to equation (1). Now obviously I have to somehow use (3.71) ,which I do know how, to derive to express the second order derivative.
On the internet I found equation (3), and I have tried to understand where this comes from (4).
Is equation (3) correct, if yes, how am i supposed to contineu from what I have in (4). It should equal 3 times the secobd order derivative?

If equation (3) is not correct, any tips how to continue from what I have?

 

[Ineedhelpimbadatphys]

The sentence after (3.71) was a little unclear. I meant that I do know how to derive (3.71) using equations (2).