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first note that $A$ only appears in the combination $M=A-BC$, so the derivative with respect to $A$ equals the derivative with respect to $M$; then write $$f(A)=||(A - BC)^Nv - w||_2^2=v^T (M^N)^TM^Nv+w^Tw-2w^TM^Nv$$ $$\Rightarrow \partial f/\partial A=Nv^T (M^N)^TM^{N-1}v+ Nv^T (M^{N-1})^TM^Nv-2Nw^TM^{N-1}v .$$ this can also be written as the inner product between two vectors $x$ and $y$: $$\partial f/\partial A=x^Ty,\;\;\text{with}\;\;y=N(A-BC)^{N-1}v,\;\;x=(A-BC)^Nv+M^TM^{N-1}v-2w.$$