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Is there a way to calculate $$\frac{\partial\; \mbox{tr}\{\log(X^tBX)\}}{\partial X},$$ where $X$ and $B$ are $n\times n$ matrices?

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$${\rm tr}\log (X^t B X)=\log{\rm det}\,(X^t BX)=2\log{\rm det}\,X+\log{\rm det}\,B$$ $$\Rightarrow\frac{\partial}{\partial X}{\rm tr}\log (X^t B X)=2X^{-1},$$ in view of Jacobi's formula

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    Yes! Thanks! There is a bit mess with transposes, but I can fix it (X is not symmetric). – 2017-02-16
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    it doesn't matter if X is symmetric or not, since the determinant of X is the same as the determinant of its transpose – 2017-02-16
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    Would be great if such questions were routed to math.SE :-) (including say the slightly more interesting version with a rectangular $X$) – 2017-02-20