I'm puzzling about how to deal with the differential of a transposed matrix. I was wondering if there is some rule such that $d(X^{T}) = (dX)^{T}$.
In general I work with derivation on the trace of a matrix and I get sometimes the following situation: $$ tr(d(X^{T})AX + Bd(X^{T})CX + DdX) $$ where X can be a rectangular matrix.
I'm quite sure that such expression can be rearranged as follows: $$ tr((AX + CXB)d(X^{T})) + tr(DdX) $$
Clearly I would like to obtain something like $tr(J(X)dX)$ for derivative, but I'm not able to go on.
Some suggestion?