I'm trying to find the gradient of the negative log-likelihood function for data following a multivariate Gaussian distribution. The negative log-likelihood function is given by $$L(\pmb{\theta}) =-\ln p(\mathcal{D}|\pmb{\theta}) = \frac{1}{2}\ln |\pmb{\Sigma}(\pmb{\theta})| + \frac{1}{2}(\pmb{y}-\pmb{\mu}(\pmb{\theta}))^{T}\pmb{\Sigma}^{-1}(\pmb{\theta})(\pmb{y}-\pmb{\mu}(\pmb{\theta})) $$ where $\pmb{y}$ and $\pmb{\mu}(\pmb{\theta})$ are $N\times1$ vectors and $\pmb{\Sigma}(\pmb{\theta})$ is the $N\times N$ covariance matrix. $\pmb{\theta}$ is the $n_{p}\times 1$ vector of parameters on which both the mean vector $\pmb{\mu}$ and the covariance matrix $\pmb{\Sigma}$ depend.
My objective is to differentiate $L(\pmb{\theta})$ with respect to $\pmb{\theta}$. I'll be using the denominator layout.
$$\frac{dL(\pmb{\theta})}{d\pmb{\theta}} = \frac{d\pmb{\Sigma}}{d\pmb{\theta}}\frac{\partial L(\pmb{\theta})}{\partial \pmb{\Sigma}} + \frac{d\pmb{\mu}}{d\pmb{\theta}}\frac{\partial L(\pmb{\theta})}{\partial \pmb{\mu}}. $$ Now, $\frac{dL(\pmb{\theta})}{d\pmb{\theta}}$ will be of size $n_{p}\times 1$. On the right hand side, $\frac{d\pmb{\mu}}{d\pmb{\theta}}$ and $\frac{\partial L(\pmb{\theta})}{\partial \pmb{\mu}}$ have sizes $n_{p}\times N$ and $N\times 1$ respectively. So, the second term has the same size as the left hand side. However, I'm unable to carry out the correct multiplication operation between the two derivatives in the first term on the right hand size.
Could anyone please help me figure out how to do this?