I need to differentiate the following equation twice with respect to $\alpha$. It is a profile log likelihood equation, where I need the derivatives to get the information matrix.
The equation is: $\widetilde{L_\beta}(\alpha)=(1-\alpha)\sum\limits_{i=1}^m n_i\log\left(w_i/\hat{\sigma}_\alpha^2\right)+\sum\limits_{i=1}^m n_i\log \kappa_\alpha (-1/y_i)-\sum\limits_{i=1}^m 1_{\lbrace n_i \neq 0 \rbrace}\log\Gamma(-n_i\alpha) + n_+(\alpha-1)$,
where
$\hat{\sigma}_\alpha^2=\frac{-\sum\limits_{i=1}^m w_i \left\lbrace y_i \mu_i^{1/(\alpha-1)}(\alpha-1)-\kappa_\alpha (\mu_i^{1/(\alpha-1)}(\alpha-1)) \right\rbrace}{n_+(1-\alpha)}$
and
$\kappa_\alpha(\theta)=\frac{\alpha-1}{\alpha}\left(\frac{\theta}{\alpha-1}\right)^\alpha$
I have tried for hours, and hours to do it, but this equation is just too hard for me, so if anyone could give me some hints how to split this up into simpler subproblems, that would be highly appreciated. Especially differentiating $\hat{\sigma}_\alpha^2$ gives me a lot of trouble.