If we have a Banach algebra $A$ then the exponential map: $$x\mapsto \sum_{n=0}^\infty \frac{x^n}{n!}$$ is well defined. Its derivative should be a map $A\to\mathrm{End}(A)$, however it is not clear to me how to actually calculate this map. For example the derivative of $x\mapsto x^2$ at point $x$ is the map $h\mapsto xh+hx$ and in general the derivative of $x^n$ will be $\sum_{k=0}^n x^k hx^{n-k}$, which prevents me from simplifying the sum.
What is the derivative of the exponential map on a Banach algebra?