second derivative $\mbox{Tr} \left(\left(X^{\frac{3}{2}}AX^{\frac{3}{2}}\right)^{\frac{1}{2}}\right)$ with respect to $X$

Question

I need to find second derivative that is : $\partial_{X}^{2} \mbox{Tr} \left(\left(X^{\frac{3}{2}}AX^{\frac{3}{2}}\right)^{\frac{1}{2}}\right)$ with respect to $X$ where both $A$ and $X$ are symmetric $nxn$ matrices. The equation (139) page 15 of matrix cook book seems useful $\frac{\partial Tr(AX)}{\partial X}=A +A^{T}-AoI$ , where $AoI$ is hadamard elementwise product. But I am confused since I have trace of squareroot. Any help is appreciated.