Problem in understanding the partial differentiation I was reading the book on Neural Network and got stuck up on this equation given below.
\begin{eqnarray}
  z^{l+1}_k = \sum_j w^{l+1}_{kj} a^l_j +b^{l+1}_k = \sum_j w^{l+1}_{kj} \sigma(z^l_j) +b^{l+1}_k.
\tag{43}\end{eqnarray}
here w and b are constant vectors. The equation was then differentiated and below is the result. Where did the summation go?
\begin{eqnarray}
  \frac{\partial z^{l+1}_k}{\partial z^l_j} = w^{l+1}_{kj} \sigma'(z^l_j).
\tag{44}\end{eqnarray}
If you want to see the complete problem then visit http://neuralnetworksanddeeplearning.com/chap2.html eqn 43 and 44
 A: When taking partial derivative w.r.t. some variable, we treat the other variables as constant. So when the expression on the right is partially differentiated w.r.t. the variable $z_j^l$, all other variables with index other than $j$ are treated constant and so $$\dfrac{\partial \sigma(z_k^l)}{\partial z_j^l}=0$$ when $j\neq k$ and $$\dfrac{\partial \sigma(z_k^l)}{\partial z_j^l}=\sigma'(z_j^l)$$ when $j=k$. I suppose this is the reason the summation vanishes from the right.
A: The $z_k$'s are defined over a sum of $z_j$'s.
$$
z^{l+1}_k = \sum_j w^{l+1}_{kj} a^l_j +b^{l+1}_k = \sum_j w^{l+1}_{kj} \sigma(z^l_j) +b^{l+1}_k.
$$
You are differentiating with respect to only one of the $z_j$'s. So, for each $j$, all the other terms in the sum go to zero: 
$$
\begin{align}
  \frac{\partial z^{l+1}_k}{\partial z^l_j} &=  \frac{\partial }{\partial z^l_j} \left[\sum_j w^{l+1}_{kj} \sigma(z^l_j) +b^{l+1}_k\right]\\
&= \sum_j \frac{\partial }{\partial z^l_j}\left[w^{l+1}_{kj} \sigma(z^l_j) +b^{l+1}_k\right]\\
&= w^{l+1}_{kj} \sigma'(z^l_j).
\tag{44}\end{align}
$$
A: I can't comment yet. It looks like either Einstein summation notation or only one term in the summation is a function of $z_j^l$. I'll say the latter seeing as it says $\sigma(z_j^l)$. $a$ must also be a constant vector.  
edit: Einstein summation just means that the sum is implied. You write the variable to be summed over, but not the summation symbol. Very useful at times.
https://en.wikipedia.org/wiki/Free_variables_and_bound_variables
https://en.wikipedia.org/wiki/Einstein_notation
