We have that (replacing \( L \) with a general layer \( l \))
$$ \delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}. $$We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have
$$ \delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, $$and recalling that
$$ z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1}, $$with \( M_l \) being the number of nodes in layer \( l \), we obtain
$$ \delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), $$This is our final equation.
We are now ready to set up the algorithm for back propagation and learning the weights and biases.