= Solution
The radial <unit vector> in the <spherical coordinate system> is
$$
\mathbf e_r=\frac{\mathbf r}{r}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).
$$
The coordinate tangent in the polar direction has length $r$, so its corresponding <unit vector> is
$$
\mathbf e_\theta=\frac1r\frac{\partial\mathbf r}{\partial\theta}
=(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta).
$$
Differentiating $\mathbf e_r$ at fixed $\phi$ therefore gives
$$
\boxed{\frac{\partial\mathbf e_r}{\partial\theta}=\mathbf e_\theta,\qquad\alpha=1.}
$$
The factor is not $r$: the corresponding <scale factors of orthogonal coordinates> occur in the <derivative> of the position <vector>, whereas the question differentiates its unit direction. The formula is understood away from the <coordinate singularities> where a complete spherical <basis> is not uniquely specified.
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