If you mean for how to use Taylor expansions and linear algebra, here's one I just made up. Let's say I want to differentiate tr(X^T X), tr is the trace, X is

If you mean for how to use Taylor expansions and linear algebra, here's one I just made up. Let's say I want to differentiate tr(X^T X), tr is the trace, X is a matrix, and X^T is its transpose. Expand: tr((X + dX)^T (X + dX)) = tr(X^T X) + 2 tr(X^T dX) + tr(dX^T dX). Our knowledge of linear algebra tells us that tr is a linear map. Hence, dX -> 2 tr(X^T dX) is the linear mapping corresponding to the Jacobian of tr(X^T X). With a little more work we could figure out how to write it as a matrix.