Let us define a function $f$ from $M(n,\mathbb{R})$ to $M(n,\mathbb{R})$ by treating $M(n,\mathbb{R})\approx\mathbb{R}^{n^2}$, by $f(X)=e^X+X$ where $e^X=1+X/{1!}+X^2/{2!}+\dots$ I want to find the (Frechet) derivative of $f$.
We know, if derivative exists at $X$, then f(X+H)-f(X)=f'(X)H+r(H) where $r(H)/\|H\|\to \bf{0}$ as $H\to \bf{0}$.
So I went on to find the difference, but couldn't figure out the linear part ($f'(X)H$) and the remainder part ($r(H)$).
Any help is appreciated.