## The derivative of the matrix exponential

Let $A$ be an $n \times n$ complex matrix and define the mapping $t \mapsto \mathsf{exp}(At)$ as usual. Prove that $\frac{d}{dt} \mathsf{exp}(At) = A \mathsf{exp}(At)$.

Using this previous exercise, we have $\frac{d}{dt} \mathsf{exp}(At) = \frac{d}{dt} \sum_k \frac{1}{k!} (At)^k$ $= A \sum_k \frac{1}{(k+1)!} (k+1) (At)^k$ $= A \sum_k \frac{1}{k!} (At)^k$ $= A \mathsf{exp}(At)$ as desired.