# matrices – Matrix derivative w.r.t. a general inverse form: \$(A^TA)^{-1/2}D(A^TA)^{-1/2}\$

I want to find derivative of matrix $$(A^TA)^{-1/2}D(A^TA)^{-1/2}$$ w.r.t. $$A_{ij}$$ where D is a diagonal matrix. Alternatively, it is okay too to have

$$frac{partial}{partial A_{ij}} a^T(A^TA)^{-1/2}D(A^TA)^{-1/2}b$$

Is there any reference for such problem? I have the matrix cookbook which gives results when $$D=I$$. But how is this general form evaluating to?

To give more information, empirical distribution of diagonal of diagonal matrix D converges to some known distribution.