Quantum Field Theory: commutator of covariant derivatives

I just started studying qtf and I dont understand the last lecture. In the lecture script a shortcut is defined. $P^j = \frac{1}{i}\partial_j - qA^j$ With this: $[P^j,P^k] = -\frac{q}{i}(\partial_j A^k - \partial_k A^j)$ I have a hard time understanding this commutator, since I would get. $[P^j,P^k] = -\frac{q}{i}(\partial_j A^k - A^k\partial_j - \partial_k A^j + A^j\partial_k)$

What do I not understand?

Take the commutator acting on a function $f$. Then $$\begin{split} [ P_i , P_j ] f &= [ - i \partial_i - q A_i , - i \partial_j - q A_j ]f \\ &= ( i \partial_i + q A_i )( i \partial_j + q A_j ) f -( i \partial_j + q A_j ) ( i \partial_i + q A_i ) f \\ &= - \partial_i \partial_j + i q A_i \partial_j \, f + i q \partial_i ( A_j f ) + q^2 A_i A_j \, f\,\, -\,\, i\, \leftrightarrow \,j \\ &= i q ( \partial_i A_j - \partial_j A_i ) f \end{split}$$