Suppose $A_0$ a connection on principal bundle $P$, pick $\xi$ an elements in $\Omega^1(adP)$ and $e^{t\xi} \in \Omega^0(AdP)$ a one-parameter group generated by $\xi$.
（edited：Suppose $G$ the correspondent Lie group of $P$, $\Omega^0(AdP)$ means the $G-$ equivairant maps from $P$ to $G$(more explicitly, any $f \in \Omega^0(AdP)$ satisfied $f(pg^{-1})=gf(p)g^{-1}$ ) and $\Omega^0(adP)$ is the section of associated vector bundle(Whose correspondent vector space is $lie(G)$ and quotient the adjoint action), which also can be viewed as lie algebra of $\Omega^0(AdP)$）
My problem is, how to show that $\frac{d}{dt}(\exp^{t\xi}•A_0)=-\nabla_{A_0}\xi$? here $g•A_0$ denotes the gauge transformation given by $g\in Aut(P),\nabla_{A_0}$ the induced exterior covariant on $\Omega^1(adP)$.
Thanks you for your answer！