Let $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$. Moreover let $\partial_{x_i} = \partial /\partial x_i$.
I would like to verify that
$$\langle x, \nabla \rangle^2 = \langle x, \nabla \rangle + \sum \limits_{1 \le i, j \le n} x_i x_j \partial_{x_i} \partial_{x_i}.$$
It this formula correct? I don't know where the first term on the RHS comes from. I would appreciate any help.
Just test it against a function: $$\langle x,\nabla\rangle^2f = \langle x,\nabla\rangle \sum_i x_i \partial_i f = \sum_{i,j} x_j\partial_j(x_i\partial_if)= \sum_i x_i \partial_if+\sum_{i,j}x_ix_j\partial_i\partial_jf,$$since $\partial_jx_i$ is $1$ if $i=j$ and zero else. More precisely: $$\sum_{i,j} x_j(\partial_jx_i)\partial_if = \sum_{i,j}x_j\delta_{ij}\partial_if = \sum_i x_i\partial_if.$$This is $\langle x,\nabla\rangle f$.
TL;DR: the formula is correct by the product rule.