From matrix cookbook (eq. 125) I only know that

$\frac{{\operatorname{Tr}((S^TS)^{-1}A)}}{dS}=-S(S^TS)^{-1}(A+A^T)(S^TS)^{-1}$.

Can anybody tell me what happens to the result if I switch $(S^TS)^{-1}$ to $(SS^T)^{-1}$?

So what is

$\frac{{\operatorname{Tr}((SS^T)^{-1}A)}}{dS}=...$?

*P.S. In general I am also very interested how the authors came up with the solution in eq. 125*