How to ("geometrically") differentiate unit vectors of spherical coordinates?

Question

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.

Answer 1

The drawing in my initial figure is wrong. The figure attached below is the correct one and it saliently provides the answer. The angle of our interest is $\angle BAC$ named $\alpha$. To find this, $$ \tan{\phi}=\frac{AB}{OB}=\frac{AB}{r}$$ $$ AB=r\tan{\phi} $$ Now, $$ \tan{\alpha}=\frac{BC}{AB}$$ From the figure it is clear that $BC=r\sin{\phi}d\theta$ and we found that $AB=r\tan{\phi}$. Substituting, $$ \tan{\alpha}=\frac{r\sin{\phi}\;d\theta}{r\tan{\phi}}=\cos{\phi}\;d\theta$$

Hence the magnitude of change in the unit vector $ \hat{\theta}$ is $1\cdot\alpha=\tan{\alpha}=\cos{\phi}\;d\theta\;$ for small angles. From the figure it is clear that the direction is $\hat{\theta}$. Putting all of this together, $$ \frac{\partial\hat{\phi}}{\partial\theta}=\cos{\phi}\;\hat{\theta}$$