I have been trying to derive the expressions of partial derivatives of unit vectors with respect to each other in the spherical coordinate system. I was able to get all of them except $\frac{\partial \hat{\phi}}{\partial\theta}$
Note: I am using the mathematics convention of denoting azimuthal angle as $\phi$ and polar angle as $\theta$
Here's the figure I've made but I'm not able to find the angle of interest marked with a question mark in the figure.