# derivation of pigou taxes in multiple polluter case

I cant quite wrap my head around one step in deriving the pigou tax in a multiple polluter case.

There are j firms, which produce joint product in amount of $$k_j$$. Joint product can be abated, where the cost function $$C_j(a_j)$$ denotes the costs for firm j for abating $$a_j$$

Social damage costs: $$e = \sum_{j=1}^{m}{(k_j -a_j) }$$

To minimize total social cost:

$$min_{(aj)_{j=1}^{m}}\sum_{j=1}^{m}C_j(a_j)+ \sum_{i=1}^{m}D_i(e)$$

With FOC we should, according to my script, arrive at:

$$C_j^(a_j)=\sum_{i=1}^{n}D_j^(e)$$ for all j = 1,...,m

But I don't understand why it isn't: $$\sum_{j=1}^{m}C_j^(a_j)=-\sum_{i=1}^{n}D_j^(e)$$

I think for example the partial derivative of $$\sum_{j=1}^{2}C_j$$ is equal to the partial derivative of it's parts. So $$[\sum_{j=1}^{2}C_j]' = C_{1}^{'}(a_1) + C_{2}^{'}(a_2)$$ and therefore:

$$\sum_{j=1}^{m}C_j^(a_j)+\sum_{i=1}^{n}D_j^(e)= 0$$

I don't really know, where Im going wrong with this and would be really happy if someone could point me into the right direction.

Best regards :)