I have a symmetric matrix $A(x)\in \mathbb{R}^{JxJ}$ indexed by $x\in \mathbb{R}^K$.
I'am interested in $$\frac{\partial (A(x)^+)^{1/2}}{\partial x_k}$$ where $M^+$ is the Moore-Pseudo-Inverse of the matrix $M$ and $M^{-1/2}$ is the matrix square root.
I'm able to compute the derivative of the usual square root inverse of $A$, i.e. $$\frac{\partial A(x)^{-1/2}}{\partial x_k}$$ via the eigenvalue decomposition and its derivatives. However, for numerical reasons I want to switch to the MPI.
I also know how to compute the derivative $$\frac{\partial A(x)^+}{\partial x_k}$$ given by $$-A^+ dA A^+ + (A^+)^2dA(I-AA^+) + (I -AA^+)dA (A^+)^2.$$
But I don't really see how I can combine these parts. I would be very grateful for any hints.