Deriving eq. (8) from (7)

[mlerma54]

Hello, I am having a hard time getting equation (8) from (7). Note that in eq. (7) index k is a summation index, hence it is better to rename it so we do not confuse it with the superidex of the activation A_{ij}^k respect to which we are taking partial derivative. After that I get equation (8'), which is different from (8). Note that in (8') there is a summation and some cross derivatives that do not appear in (8).

The next partial derivative does not produce (9) either, and it does not seem clear how to isolate the \alpha_{ij}^{kc} this way.

Do you know what is wrong? Thank you.