Reference for a derivative formula for matrices I found the following identity:
$$ \frac{\partial( \det (X^T A X ))}{\partial X} = 2\det(X^TAX)AX(X^TAX)^{-1} $$
on the matrix cookbook.  It is equation 47 on page 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?

NOTE:  I asked this question on Mathoverflow and was advised to repost it here.
 A: Of course we must assume $X^T A X$ is invertible for this equation to make sense.
Take $\tilde{X} = X + t Y$.  Then 
$$\eqalign{\tilde{X}^T A \tilde{X} &= X^T A X + t (Y^T A X + X^T A Y) + O(t^2)\cr
 &= \left(I + t (Y^T A X + X^T A Y)(X^T A X)^{-1}\right) X^T A X + O(t^2)\cr}$$
so
$ \det(\tilde{X}^T A \tilde{X}) = \det(I + tM) \det(X^T A X) + O(t^2)$ where $M = (Y^T A X + X^T A Y)(X^T A X)^{-1}$.
Now $\det(I+tM) = \det(\exp(tM)) + O(t^2) = \exp(\text{tr}(tM)) + O(t^2) = 1 + t\; \text{tr}(M) + O(t^2)$.  We have
$$ \eqalign{\text{tr}(M) &= \text{tr}(Y^T A X (X^T A X)^{-1} + X^T A Y (X^T A X)^{-1})\cr
&= \text{tr}(Y^T A X (X^T A X)^{-1}) + \text{tr}((X^T A X)^{-1} X^T A Y)\cr
&= 2 \text{tr}(Y^T A X (X^T A X)^{-1})}$$
Taking $Y = E_{ij}$, the matrix with $1$ in entry $(i,j)$ and $0$ everywhere else, 
this says 
$$ \frac{\partial}{\partial X_{ij}} \det(X^T A X) = 2 \text{tr}(E_{ji} A X (X^TAX)^{-1}) \det(X^T A X) = 2 \det(X^T A X) (A X (X^T A X)^{-1})_{ij} $$
which is the meaning of
$$ \frac{\partial}{\partial X} \det(X^T A X) = 2 \det(X^T A X) A X (X^T A X)^{-1}  $$
A: The requested formula is absolutely incorrect. 
Let $\phi:X\rightarrow X^TAX=U\rightarrow \det(X^TAX)$. The hypothesis $X^TAX$ invertible is useless. 
$U'_X:H\rightarrow 2X^TAH$ and 
$\phi'(X):H\rightarrow tr(adj(U)U'(H))$ (where $adj(U)$ is the classical adjoint of $U$).
Finally $\phi'(X)(H)=2tr(adj(X^TAX)X^TAH)$.
In particular $\phi'(X)=0$ iff $adj(X^TAX)X^TA=0$.
