= Solution
Differentiating the <spherical coordinate system>
$$
\mathbf x=(r\sin\theta\cos\phi,\,
r\sin\theta\sin\phi,\,
r\cos\theta)
$$
gives the orthogonal decomposition
$$
d\mathbf x
=\mathbf e_r\,dr+r\mathbf e_\theta\,d\theta
+r\sin\theta\,\mathbf e_\phi\,d\phi,
$$
where
$$
\begin{aligned}
\mathbf e_r&=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),\\
\mathbf e_\theta&=(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta),\\
\mathbf e_\phi&=(-\sin\phi,\cos\phi,0).
\end{aligned}
$$
Thus the <scale factors of orthogonal coordinates> are
$$
\boxed{h_r=1,\qquad h_\theta=r,\qquad h_\phi=r\sin\theta}.
$$
Since the <total differential> satisfies
$$
df=f_r\,dr+f_\theta\,d\theta+f_\phi\,d\phi
=d\mathbf x\cdot\nabla f,
$$
comparison of coefficients gives the <gradient>
$$
\boxed{
\nabla f
=\mathbf e_r\frac{\partial f}{\partial r}
+\mathbf e_\theta\frac1r\frac{\partial f}{\partial\theta}
+\mathbf e_\phi\frac1{r\sin\theta}\frac{\partial f}{\partial\phi}}.
$$
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