Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that $\varphi_\alpha$ is twice differentiable at 0 and that $$\varphi_\alpha''(0) = \operatorname{det}^{-\alpha}(A)\left(\alpha \operatorname{Tr}^2(A^{-1}M) + \operatorname{Tr}\left((A^{-1}M)^2\right)\right).$$
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that $\varphi_\alpha$ is twice differentiable at 0 and that
$$\varphi_\alpha''(0) = \operatorname{det}^{-\alpha}(A)\left(\alpha \operatorname{Tr}^2(A^{-1}M) + \operatorname{Tr}\left((A^{-1}M)^2\right)\right).$$