The problem is stated in the title. Here, $A_{ij}$ denotes element $ij$ in the adjugate matrix of $A$, and $a_{ij}$ denotes element $ij$ in matrix $A$. I did the following
$$ \frac{\partial}{\partial a_{ij}} \det A = \frac{\partial}{\partial a_{ij}} \left( \sum_i a_{ij} A_{ij} \right) = \sum_i A_{ij},$$
which is not what I was supposed to prove. Where is my argument wrong and how would I prove this statement? Thanks!
Your first equality was wrong because of a conflict of notations on $i$.
$$ \frac{\partial(\det A)}{\partial a_{ij}}= \frac{\partial(\sum_ka_{kj}A_{kj})}{\partial a_{ij}}=A_{ij}. $$