$\mathbf{15}$ ▷ Let $M \in \mathcal{M}_n(\mathbf{R})$, which can also be considered as a complex matrix, and let the application $\delta_M : \mathbf{R} \rightarrow \mathbf{R},\ t \mapsto \delta_M(t) = \operatorname{det}\left(I_n + tM\right)$. Using a Taylor expansion to order 1, show that $\delta_M$ is differentiable at 0 and compute $\delta_M'(0)$.