This is related to another question I have asked but I'm interested in the derivative of $|S|$ with respect to $S$. Here, $S$ is a symmetric, non-singular matrix and $|S|$ is the absolute value function i.e. Let $S = QDQ^T$ be an eigenvalue decomposition of $S$. Then we define $|S| = Q|D|Q^T$ as the matrix with the same eigenvectors as $S$ but whose eigenvalues are the absolute values of the respective eigenvalues of $S$.
A related problem is the derivative of $\text{sign}(S)$ with respect to $S$. Where, this time, $\text{sign}(S) = Q \text{sign}(D)Q^T$ has eigenvalues equal to the sign of the respective eigenvalues of $S$. The relation comes from the fact that $|S| = S\text{sign}(S) = \text{sign}(S)S$.
Since $S$ has no zero eigenvalues, I'd like to think the derivative of $|S|$ and $\text{sign}(S)$ exist but I'm struggling to see how to move forwards.