My math are a little bit rusty, I'm trying to reproduce the derivative of the below equation w.r.t α from this paper (equation 4.9, result on 4.10).
The authors estimate α by zeroing the first derivative of the right-hand side of inequality. It would be really helpful to see the intermediate steps.
Let $S_1 = \sum_iD^t(i)\frac{1-C_iy_ih_t(x_i)}{2}$ and $S_2 = \sum_iD^t(i)\frac{1+C_iy_ih_t(x_i)}{2}$. Then, the right expresion is $$ f(\alpha) = e^{\alpha}S_1+e^{-\alpha}S_2. $$ Therefore, $$ f'(\alpha) = e^{\alpha}S_1-e^{-\alpha}S_2. $$ Making $f'(\alpha)=0$ one get $\alpha+\log S_1=-\alpha +\log S_2$ and $\alpha=(1/2)\log(S_2/S_1)$.