I'm trying to understand the derivation of the log-likelihood function for Gaussian Mixture Models. According to my records the following steps are made.
The log-likelihood function is defined as:
$ L(X|\Theta) = \sum_{n=1}^N ln P(x_n|\Theta) = \sum_{n=1}^N ln \sum^M_{m=1} \alpha_m \mathcal{N}(x_n|\mu_m, \Sigma_m) $
And then the partial derivative w.r.t. $\mu_m$ is made, where the first step seems fine for me, applying the chain rule and the derivative of the log:
$$ \frac{\partial ln P(X|\theta)}{\partial \mu_m} = \sum_{n=1}^N \frac{1}{\sum_{m'=1}^M\alpha_{m'} \mathcal{N}(x_n|\mu_{m'},\Sigma_{m'})} \cdot \frac{\partial \sum^M_{m=1} \alpha_m \mathcal{N}(x_n|\mu_m, \Sigma_m)}{\partial\mu_m} $$
I just don't get how the prior equation leads to the following:
$$ \frac{\partial ln P(X|\theta)}{\partial \mu_m} = \sum_{n=1}^N \frac{\alpha_m \mathcal{N}(x_n|\mu_n, \Sigma_m)}{\sum_{m'=1}^M\alpha_{m'} \mathcal{N}(x_n|\mu_{m'},\Sigma_{m'})} \cdot \frac{\partial[ln(\alpha_m) + ln \mathcal{N}(x_n|\mu_m, \Sigma_m)]}{\partial\mu_m} $$
From where does the log arises here again after the derivation?