Solution
= Solution
Since $r=(x_jx_j)^{1/2}$, the <chain rule> gives
$$
\boxed{\frac{\partial r}{\partial x_i}=\frac{x_i}{r}}.
$$
For the radial <vector field> $g(r)\mathbf x$, the <product rule> and <Einstein summation convention> give
$$
\begin{aligned}
\nabla\cdot\bigl(g(r)\mathbf x\bigr)
&=\frac{\partial}{\partial x_i}\bigl(g(r)x_i\bigr)\\
&=g'(r)\frac{x_i}{r}x_i+3g(r)\\
&=\boxed{rg'(r)+3g(r)}.
\end{aligned}
$$