The proof of $\frac{\partial g_j}{\partial x_i} g_i = \frac{\partial}{\partial x_i} \Big(\frac{1}{2} |\vec{g}|^2 \delta_{ij} \Big) $.

Question

I want to prove that

$$\frac{\partial g_j}{\partial x_i} g_i = \frac{\partial}{\partial x_i} \Big(\frac{1}{2} |\vec{g}|^2 \delta_{ij} \Big) $$ where $g_i$ is from the gravitational force $F_i = m g_i$.

Could you tell me how the Kronecker-Delta symbol come into this equation?