In Rossman (Lie Groups - An introduction through linear groups), he makes the following statement:
Theorem: $$\exp'_X(Y)=\exp(X)\frac{1-\exp(-ad_X)}{ad_X} Y,$$
where $$\frac{1-\exp(-ad_X)}{ad_X}=\sum_{k=0}^{\infty} \frac{(-1)^k}{(k+1)!} (ad_X)^k,$$ and $ad_X(Y)=XY-YX$.
Now, the way he proves the theorem is by manipulating a differential equation. However, by directly computing the derivative of $\exp$, we arrive at:
$$\exp'_X(H)=H+\frac{XH+HX}{2}+\frac{X^2H+XHX+HX^2}{3!} +\cdots$$
I can't help looking at the above equation and trying to arrive at the theorem through it alone by manipulating the terms of $\exp(-X)$ times the above series, but I can't seem to handle them properly. Is it possible to do this in a clean way?