I need to show that $\displaystyle \nabla \times (f F)= f \nabla \times F+ (\nabla f) \times F$

How will I represent the scalar function? Do I write $\displaystyle f=\psi(x,y,z)$ or

$\displaystyle f=A_x+A_y+A_z$

I chose $\displaystyle F=a_x \vec i +a_y \vec j +a_z \vec k$

Using $\displaystyle f=\psi(x,y,z)$ I work out the LHS of question as

$\displaystyle (\psi a_z)_y-(\psi a_y)_z-(\psi a_z)_x+(\psi a_x)_z+(\psi a_y)_x-(\psi a_x)_y$.............How to go further?