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 $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$. We define the application $\varphi_\alpha$ by $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM).$$ Show that $\varphi_\alpha$ is differentiable on $]-\varepsilon_0, \varepsilon_0[$ and that $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha'(t) = -\operatorname{Tr}\left((A + tM)^{-1}M\right) \operatorname{det}^{-\alpha}(A + tM).$$
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 $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$. We define the application $\varphi_\alpha$ by
$$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM).$$
Show that $\varphi_\alpha$ is differentiable on $]-\varepsilon_0, \varepsilon_0[$ and that
$$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha'(t) = -\operatorname{Tr}\left((A + tM)^{-1}M\right) \operatorname{det}^{-\alpha}(A + tM).$$