I am trying to show that $$\sum_{j=1}^3 x_j \frac{\partial}{\partial x_j} = r \frac{\partial}{\partial r}.$$ I have $$\frac{\partial f}{\partial x_j} = \frac{\partial f}{\partial r} \frac{\partial r}{\partial x_j} = \frac{\partial f}{\partial r} \frac{x_i}{r},$$ for any function $f,$ which implies that $$x_i r^2 \frac{\partial f}{\partial x_j} = x_i^2 r\frac{\partial f}{\partial r}.$$ Summing over $i$ gives the result.
But is there a better proof of this fact?