Update of sums of Gasussian Mixture Model

This question is motivated by this post. Then the log multivariate normal density looks like this:

$$$$\implies \log(1)+ \frac{p}{2}\log(2\pi)+ \frac{1}{2}\log(|\Sigma^X_j+\Sigma^Y_{k}|^{-1})- \frac{1}{2}(Z_i-\mu^X_j)^T(\Sigma^X_j+\Sigma^Y_k)^{-1}(Z_i-\mu^X_j)$$$$

Now, One way is to use EM to update the estimates, but in M-step, one could encounter taking derivative of $$\log(det(\Sigma^X_j+\Sigma^Y_{k})^{-1})$$ or derivative of above equation respect to $$\Sigma^X_j$$ or $$\Sigma^Y_k$$, how can this be done? Many thanks in advance.