If you just crunch it out by components you can see that it works out.
$$\begin{align}\vec\nabla\times(f\vec G)&=\left\langle\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right\rangle\times\left\langle fG_x,fG_y,fG_z\right\rangle\\
&=\left\langle\frac{\partial(fG_z)}{\partial y}-\frac{\partial(fG_y)}{\partial z},\frac{\partial(fG_x)}{\partial z}-\frac{\partial(fG_z)}{\partial x},\frac{\partial(fG_y)}{\partial x}-\frac{\partial(fG_x)}{\partial y}\right\rangle\\
&=\left\langle\frac{\partial f}{\partial y}G_z-\frac{\partial f}{\partial z}G_y,\frac{\partial f}{\partial z}G_x-\frac{\partial f}{\partial x}G_z,\frac{\partial f}{\partial x}G_y-\frac{\partial f}{\partial y}G_x\right\rangle\\
&\quad+f\left\langle\frac{\partial G_z}{\partial y}-\frac{\partial G_y}{\partial z},\frac{\partial G_x}{\partial z}-\frac{\partial G_z}{\partial x},\frac{\partial G_y}{\partial x}-\frac{\partial G_x}{\partial y}\right\rangle\\
&=\left\langle\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right\rangle\times\left\langle G_x,G_y,G_z\right\rangle+f\left\langle\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right\rangle\times\left\langle G_x,G_y,G_z\right\rangle\\
&=\left(\vec\nabla f\right)\times\vec G+f\vec\nabla\times\vec G\end{align}$$
I'm not sure about the product rule notation you are referring to is. Can you clarify a bit?