Doubt in Derivation of Backpropagation
For a while I thought you were right, but the equation is correct. 

If you think about it conceptually,  then I think you will see that it has to be correct. When updating $w_{ij}$, you have to make the change that improves the total error $E$, and not just the error $E_k$ of output node $k$, right? So you have to be looking for the weights that give you the lowest $E$, that is, where $\partial E / \partial a_k$ is 0.

If you write out the sum over $k$ in the equation for $E$, you get one term for $a_1$, one for $a_2$, and so on. The sum over $k$ in the equation that confuses you, just makes sure to first get the $a_1$ term from $E$, then the $a_2$ term, and so in. It first differentiates $E$ to $a_1$, which is $t_1 + a_1$ if I did that correctly, and gets the correct tion for that. Then for $a_2$, until the end, and finally you have to sum over $k$ to add up the corrections from all terms.