Can you please help if I'm calculating this correctly? $$\frac {\delta} {\delta \beta} [\text{ln}(g^{-1}(\alpha + \beta x_i))] = \frac {x_i (g'(g^{-1}(\alpha + \beta x_i)))} {g^{-1}(\alpha + \beta x_i)} $$
I'm not sure if this can be simplified further.
I also need help in calculating the second derivative with respect to $\beta$
It's incorrect. From the differentiation rules $$ [f^{-1}(x)]'=\{f'[f^{-1}(x)]\}^{-1} $$ and $$ \left[\ln(f(x))\right]'=\frac{f'(x)}{f(x)} $$
we have
$$ D\equiv\frac{\partial}{\partial \beta}\{\ln[f^{-1}(g(\beta))]\}=\frac{[f^{-1}(g(\beta))]'}{f^{-1}(g(\beta))}=\frac{\{f'[f^{-1}(g(\beta))]\}^{-1}g'(\beta)}{f^{-1}(g(\beta))} $$
so
$$ D=\frac{1}{f'[f^{-1}(g(\beta))]}\frac{g'(\beta)}{f^{-1}(g(\beta))} $$
if $g(\beta)=\alpha + \beta x_i$ then
$$ \frac{\partial}{\partial \beta}\{\ln[f^{-1}(\alpha + \beta x_i)]\}=\frac{1}{f'[f^{-1}(\alpha + \beta x_i)]}\frac{x_i}{f^{-1}(\alpha + \beta x_i)} $$