Somehow I've gotten myself confused trying to take the derivative of the exponential map on $\mathfrak{so(3)}$.
For vector $\theta$, $\delta \theta$, and $p \in \mathbb{R}^3$, define $$R(\theta, p) = e^{[\theta]_\times}p$$ as the map that rotates $p$ about the axis $\theta$ by $\|\theta\|$ radians counterclockwise.
I now want to calculate the derivative $D_\theta R(\theta_0, p)$. We know that $D_\theta R(0,p) \delta\theta = [\delta\theta]_\times p,$ so I thought it would follow that
\begin{align*} D_\theta R(\theta_0,p) &= e^{[\theta_0]_\times}D_\theta R(0,p)e^{-[\theta_0]_\times}\delta\theta\\ &= e^{[\theta_0]_\times}\left[e^{-[\theta_0]_\times}\delta\theta\right]_\times p\\ &= -e^{[\theta_0]_\times}\left[p\right]_\times e^{-[\theta_0]_\times}\delta\theta \end{align*} and hence $$D_\theta R(\theta_0, p) = -e^{[\theta_0]_\times}\left[p\right]_\times e^{-[\theta_0]_\times},$$ but trying some numerical examples I see this is not correct. Where have I gone wrong?