I found the identity

$$ \frac{\partial( \det (X^T A X ))}{\partial X} = 2\det(X^TAX)AX(X^TAX)^{-1} $$

On the matrix cookbook (http://orion.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf). It is equation 47 on p. 8. Note that $X$ is an $n \times m$ matrix and $A$ is a symmetric $n \times n$ matrix.

I could not find the identity in their cited references...does anyone know of a textbook or paper that has this identity?