I am studying the Drinfel'd-Sokolov hierarchies. And here is what confuses me. Let $\mathfrak{g}$ be a (simple) finite-dimensional Lie algebra, let $u(x), v(x) \in C^\infty(\mathbb{R},\mathfrak{g})$. We can define the exponential of $F\in\operatorname{End}(\mathfrak{g})$ as

$\exp(F) := \mathrm{id} + F + \frac{1}{2!}(F\circ F)+\cdots \in \operatorname{GL}(\mathfrak{g})$

**Question**: how to calculate $\frac{d}{dx} \exp(\operatorname{ad} u(x))(v(x))$?

I believe there is an explicit formula but after long time searching, I cannot find any. And I hope I can get some inspiration here.