What is the derivative of general 3D rotation with respect to one angular component?

Question

For a general rotation $R(t_1, t_2, t_3)$ where the $t_i$'s are the components of the rotation vector in the axis-angle representation. Is there closed formula for the derivative of $dR/dt_i$?

I only find derivative with respect to the magnitude of the angle, but not to the components of $t_i$

$$ \frac {dR(\theta)}{d\theta}=L_\theta R(\theta) $$ where $\theta=norm([t_1, t_2, t_3])$ is the rotation angle, and $L_\theta$ is the generator

https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#:~:text=In%20mathematics%2C%20the%20axis%E2%80%93angle,the%20rotation%20about%20the%20axis.