# Derivation of M-step in EM algorithm for mixture of Gaussians

I am trying to derive the parameter estimation equations for the M-step of the expectation maximization (EM) algorithm for a mixture Gaussians when all Gaussians share the same covariance matrix $\mathbf{\Sigma}$.

Pattern Recognition and Machine Learning by Bishop has a section on EM for Gaussian mixtures, and it includes a derivation of the M-step when all $K$ Gaussians have different covariance matrices $\mathbf{\Sigma_k}$. I think that if I can understand this derivation well, I can modify it to get what I want.

I understand the derivation given by Bishop for the M-step equation for $\mathbf{\mu_k}$. However, the book does not show detailed steps for the derivation of the M-step for $\mathbf{\Sigma_k}$. When I tried to derive it myself by computing $\frac{\partial \mathbf{L}}{\partial \mathbf{\Sigma_k}}$ and setting it to 0, I came across the following derivative that I don't know how to deal with:

$$\frac{\partial}{\partial \mathbf{\Sigma_k}} \left ( (2\pi)^{-d/2}|\mathbf{\Sigma_k}|^{-1/2}e^{-\frac{1}{2}(x-\mathbf{\mu_k})^T\mathbf{\Sigma_k}^{-1}(x-\mathbf{\mu_k})}\right )$$

Basically, it's the derivative of the multivariate Gaussian pdf with respect to the covariance matrix. How do I compute this derivative? I've computed the derivative of the logarithm of this function before when studying Gaussian Bayes classifiers, so that makes me think I've made a mistake somewhere.

$$\frac{\partial \ln (f)}{\partial \mathbf{\Sigma}_k} = \frac{1}{f} \frac{\partial f}{\partial \mathbf{\Sigma}_k}\\ \Rightarrow \frac{\partial f}{\partial \mathbf{\Sigma}_k} = f \cdot\frac{\partial \ln (f)}{\partial \mathbf{\Sigma}_k}$$
Also, it turns out that taking the derivative of the PDF with respect to $\mathbf{\Sigma}^{-1}$ is easier and leads to the same answer.