Solution
= Solution
For the <operator norm>, <submultiplicativity> bounds the remainder of the <matrix exponential> by
$$
\|\exp A-I-A\|\leq\sum_{j=2}^\infty\frac{\|A\|^j}{j!}=e^{\|A\|}-1-\|A\|=O(\|A\|^2).
$$
Therefore $\|\exp A-\exp0-A\|/\|A\|\to0$. This is precisely <Fréchet differentiability> at zero, with
$$
\boxed{D\exp(0)[H]=H}.
$$