Trouble understanding derivation of covariant derivative

[bibalasvegas]

Hi, I'm having problems following a derivation for the covariant derivative. I've shown the line where I'm having trouble:

http://img15.imageshack.us/img15/49/covariantderivative.jpg

The general argument being used is that if the covariant derivative must follow the product rule it can be expressed as the partial derivative plus some linear transformation - (the left and right parts of the right side of the first line respectively). Also we are trying to find the transformation properties of the christoffel symbol by using the fact that the covariant derivative should transform as a (1,1) tensor hence the primed indices.

My problem is that I don't see how the second line follows from the first. I know they are trying to express the previous line in terms of its transformation but I thought the first term on the right hand side of the top equation would correspond only to the first term on the right hand side of the bottom equation. I don't see where the middle term of the bottom equation comes from! No doubt this is due to my lack of understanding of vector derivatives. Hope all this makes sense and any help would be greatly appreciated!

 

[bibalasvegas]

Just to say I've resolved this issue - it is solved. I wasn't acting on the whole expression for the transformed V with the partial derivative.