In the standard derivation of Jacobi's formula One uses the Laplace expansion of the determinant of some matrix $A$ : $$\text{det}\,A=\sum_{j}A_{ij}C_{ij}$$ where $i$ is a fixed, but arbitrary row, and $C_{ij}=(-1)^{i+j}M_{ij}$ are the elements of the cofactor matrix, with $M_{ij}$ the minors of $A$.
One then takes the derivative of the determinant with respect to an arbitrary element of $A$, $A_{ij}$. In every example of the derivation that I've seen the standard assumption is that, since the row over which one expands the determinant in arbitrary, one can always choose the element that we are taking the derivative with respect to such that it is along the same row, leading to $$\frac{\partial\text{det}\,A}{\partial A_{ij}}=\sum_{k}\delta_{jk}C_{ik}+\sum_{k}A_{ik}\frac{\partial C_{ik}}{\partial A_{ij}}=C_{ij}$$ where one uses the fact that none of the elements $C_{ik}$ can depend on the element $A_{ij}$ (since they are determined by deleting the $i^{th}$ row and $k^{th}$ column).
My question is, what if we take the derivative with respect to a different element, say $A_{lm}$? Is the point that whichever element of $A$ that we choose to differentiate $\text{det}\,A$ with respect to, we can always choose to expand the determinant along that row such that we end up with $\frac{\partial\text{det}\,A}{\partial A_{ij}}=C_{ij}$ in all cases? i.e. If one chooses $A_{lm}$ to differentiate $\text{det}\,A$ with respect to, then one can choose to Laplace expand $\text{det}\,A$ along its $l^{th}$ row such that $\frac{\partial\text{det}\,A}{\partial A_{lm}}=C_{lm}$.
Apologies if this is a stupid question, I just really wanted to get the concept sorted in my mind.