Define a function $P:\mathbb S^{n\times n}\rightarrow \mathbb S^{n\times n}$ and a function $Q:\mathbb S^{n\times n}\rightarrow \mathbb S^{n\times n}$. We have a linear operator $L$ such that $$ LP=Q $$ Assume the linear operator is invertible, then $$ P=L^{-1}Q $$ Define the partial derivative operator $\partial_a=\frac{\partial }{\partial a}$.
Is the following equation correct (or need any more conditions)? $$ \partial_a P=L^{-1}\partial_a LL^{-1} Q-L^{-1}\partial_aQ $$
Thanks for any ideas!
Is $L$ constant? It doesn't take a lot to convince yourself that the proof of the Leibniz rule generalises to any ring structure, so $\partial_a Q=\partial_a (LP)=(\partial_a L)P+L(\partial_a P).$ So you see the commutator of $\partial_a$ and multiplication by $L$ is exactly multiplication by $\partial_a L$, so no, the operators don't commute, unless $L$ is constant.
Of course, $L L^{-1}Q=Q,$ so $\partial_a (LL^{-1}Q)=\partial_a Q,$ automatically.