Assume that our data is distributed according to a $\underline d$ dimensional multivariate Gaussian with $\bar \mu$ mean and $\Sigma$ covariance matrix: $$(\mathbf x_1, \dots, \mathbf x_n) \sim \mathcal N(\bar \mu, \Sigma).$$
Using The Matrix Cookbook Equation [57, 59], derive $\frac{\partial \mathcal L(\theta)}{\partial \Sigma}$ in matrix form and set it to zero to find $\Sigma_{ML}$ . Assume each $x_i$ is drawn independently.