How can the following identity be proven?
$$\nabla(\vec{A} \cdot \vec{B}) = \vec{A} \times(\nabla \times \vec{B}) + \vec{B} \times(\nabla \times \vec{A}) + (\vec{A}\cdot \nabla)\vec{B} + (\vec{B} \cdot \nabla)\vec{A}$$
How can the following identity be proven?
$$\nabla(\vec{A} \cdot \vec{B}) = \vec{A} \times(\nabla \times \vec{B}) + \vec{B} \times(\nabla \times \vec{A}) + (\vec{A}\cdot \nabla)\vec{B} + (\vec{B} \cdot \nabla)\vec{A}$$