Let $Z(\alpha)=\sum_xQ(x)^\alpha$ and $\ell(x)=\log_2Q(x)$. Then
$$
H(Q_\alpha)=-\alpha\mathbb E_\alpha\ell+\log_2Z(\alpha).
$$
Differentiation of this <exponential family> gives
$$
\frac d{d\alpha}\mathbb E_\alpha\ell
=(\ln2)\operatorname{Var}_\alpha(\ell),
\qquad
\frac d{d\alpha}\log_2Z=\mathbb E_\alpha\ell.
$$
The first-order terms cancel, leaving
$$
\frac d{d\alpha}H(X_\alpha)
=-\alpha(\log_e2)
\operatorname{Var}(\log_2Q(X_\alpha)).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 224 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-224/4/b/solution