#tex{{ \nabla\cdot(f\bm{A}) &= \left(\frac{\partial}{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\cdot(fA_x,fA_y,fA_z)\\ &= \left(\frac{\partial f}{\partial x}A_x+f\frac{\partial A_x}\left(\frac{\partial f}{\partial z}A_z+f\frac{\partial A_z}{\partial z}\right)\\ &= \left(\frac{\partial f}{\partial x}A_x+\frac{\partial f}{x}+f\frac{\partial A_y}{\partial y}+f\frac{\partial A_z}{\partial z}\right)\\ &= \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y \frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\cdot(A_x,A_y,A_z)\\ &=\nabla f\cdot\bm{A}+f\nabla\cdot\bm{A} }}