Problem in gradient operator and Kronecker delta function

Question

I have this expression

$$\nabla_{i}\nabla_{j}\Big(\frac{1}{r}\Big)$$ Where $r$ is a distance. I tried this, but encountering manipulations of $\delta_{ij}$ with $\hat{r_i},\hat{r_{j}}$ and still stuck in here. The answer has this form:$$-\frac{(\delta_{ij}-3\hat{r_i}\hat{r_{j}})}{r^3}$$

Answer 1

\begin{align*} \nabla(\nabla(1/r)) & = \nabla(-(1/r^3) x) \\ (\nabla(\nabla(1/r)))_{ij} & = -\frac{\partial (x_i/r^3)}{\partial x_j} \\ & = -\frac{\frac{\partial x_i}{\partial x_j} r^3 - \frac{\partial r^3}{\partial x_j} x_i}{r^6} \\ & = -\frac{\delta_{ij}}{r^3} + \frac{\frac{\partial r^3}{\partial x_j} x_i}{r^6} \\ & = -\frac{\delta_{ij}}{r^3} + \frac{3r^2 \frac{\partial r}{\partial x_j} x_i}{r^6} \end{align*}

I think you can finish from here (since we already calculated $\frac{\partial r}{\partial x_j}$ elsewhere).