Investigating the Meaning of a Function to the Right of the Del Operator

Let $F(x,y,z) \equiv f(x) g(y) h(z)$. In this previous post, $\vec{\nabla}F(x,y,z)$ was written in terms of spherical coordinates and unit vectors for a spherical coordinate system. The corresponding equation was found to be

$ \vec{\nabla}F(x,y,z) = \vec{e}_r \frac{\partial F(x,y,z) }{\partial r} + \vec{e}_\theta \frac{1}{r} \frac{\partial F(x,y,z) }{\partial \theta} + \vec{e}_\phi \frac{1}{r\sin\theta} \frac{\partial F(x,y,z) }{\partial \phi} $.

This is (6) in Reference 2, except that I have a function $F(x,y,z)$ to the right of $\vec{\nabla}$. This equation resulted from substituting known quantities into the definition of del in terms of Cartesian coordinates and Cartesian unit vectors,

$\vec{\nabla} := \vec{e}_x\frac{\partial}{\partial x} + \vec{e}_y\frac{\partial}{\partial y} + \vec{e}_z\frac{\partial}{\partial z}$

Notice that this definition does not include a function to the right of the derivatives. It was in this post that the function $F(x,y,z) \equiv f(x) g(y) h(z)$ was first introduced. The reason for introducing this function is somewhat complicated. In particular, $F(x,y,z) \equiv f(x) g(y) h(z)$ was introduced so that the product rule could be used, to expand $\frac{\partial F(x,y,z)}{\partial r}$. The same technique of using the product rule was used for the the polar coordinate and for the azimuthal coordinate. While these posts are quite redundant, I found it useful to go through the exercise of seeing how the product rule emerged for each of the three cases. There was one case for each spherical coordinate.

I would like to investigate how the results of each of these three posts would have changed if the function $F(x,y,z)$ had not been placed to the right of a particular partial derivative for a spherical coordinate. This is important because eventually I would like to find out if it is valid to write the del operator in terms of spherical coordinates, even if there is no function such as $F(x,y,z)$ to the right of $\vec{\nabla}$.