This post is about finding the directional derivative: $$ \left . \frac{d}{d\lambda} \exp([\omega + \lambda w]_\times)\right |_{\lambda=0}, \qquad \omega, w \in \mathbb{R}^3. $$ This is related to computing the Jacobian of the exponential map taking $\mathfrak{so}(3)$ to $SO(3)$, with a particular basis chosen for $\mathfrak{so(3)}$. We will show that:

$$ \left . \frac{d}{d\lambda} \exp([\omega + \lambda w]_\times)\right |_{\lambda=0} = \exp([\omega]_\times)[D w]_\times $$ where: $$ D = \frac{\sin(\theta)}{\theta} I - \frac{1- \cos(\theta)}{\theta^2} [\omega]_\times + \frac{\theta - \sin(\theta)}{\theta^3} \omega \omega^T, \qquad \theta = \lVert \omega \rVert. $$

Proof: From the previous post on the directional derivative of the matrix exponential and the relationship between $[\cdot]_\times$ and rotation matrices discussed in another post we have: $$ \begin{align} \left . \frac{d}{d\lambda} \exp([\omega + \lambda w]_\times)\right |_{\lambda=0} &= \left . \frac{d}{d\lambda} \exp([\omega]_\times + \lambda [w]_\times) \right |_{\lambda=0}\\
&= \exp([\omega]_\times)\left(\int_0^1 \exp(-s[\omega]_\times) [w]_\times \exp(s[\omega])_\times \; ds\right)\\
&= \exp([\omega]_\times)\left(\int_0^1 [\exp(-s[\omega]_\times)w]_\times\; ds\right), \\
&= \exp([\omega]_\times)\left[\left(\int_0^1 \exp(-s[\omega]_\times) \; ds \right ) w\right ]_\times. \end{align} $$ The remaining integral is of the form discussed in yet another post, and in the notation of that post we have: $$ \begin{align} \int_0^1 \exp(-s [\omega]_\times) \; ds &= C_1(-\omega, 1) \\
&=f_1(\| \omega\|) I + f_2(\| \omega\|) [-\omega]_\times + f_3(\|\omega\|) \omega \omega^T \end{align} $$ Applying the definitions of $f_1, f_2, f_3$ from that post we have our result.