I am currently going through some derivation I made for computing an error term used in neural network, and bit confused on what happened to this summation, at the derivation?
\begin{equation} \label{eq:error_term_node_layer_output} \begin{split} c &= \frac{1}{2}||{y - a_{i}^{(l)}}||^2 \\ \delta_{i}^{(l)} &= \frac{\partial c}{\partial z_i^{(l)}} \\ & = \frac{\partial c}{\partial z_{i}^{(l+1)}} \frac{\partial z_{i}^{(l+1)}}{\partial z_i^{(l)}} \\ z_i^{(l+1)} & = \sum_{j=1}^{\text{nodes in layer} ~\text{l}} w_{ij}^{(l)} a_{i}^{(l)} + b_i^{(l)}\\ \frac{\partial c}{\partial z_{i}^{(l+1)}} &= \delta_{j}^{(l+1)}\\ %&= \sum_{j=1}^{s_{l+1}} w_{ji}^{(l)} x_{j}^{l} + b^{(l)}\\ \frac{\partial z_i^{(l+1)}}{\partial a_i^{(l)}} &= \frac{ \partial }{ \partial z^{(l)}_i } \sum_j W_{ij}^{(l)} f(z_i^{(l)}) + b_i^{(\ell)} \\ &= W_{ij}^{(l)} f'(z_i^{(l)})\\ \delta_i^{(l)} &= \delta_{j}^{(l+1)}W_{ij}^{(l)} f'(z_i^{(l)}) %\delta_i^{(l)} &= \sum_{j=1}^{s_{(l+1)}}\delta_j^{(l+1)}w_{ji}^{(l)} %\delta_i^{(l)} &= f'(z_i^{(l)})\sum_{j=1}^{s_{(l+1)}}\delta_j^{(l+1)}w_{ji}^{(l)} \end{split} \end{equation}
The math is inspired from this http://bigtheta.io/2016/02/27/the-math-behind-backpropagation.html