Math Div expression help

Is the following true?

$$\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } \nabla \cdot \vec{r}f(r) = \left ( \pd{f}{r}{} \pd{r}{x}{} \cdot x + f \right) + \left ( \pd{f}{r}{} \pd{r}{y}{} \cdot y + f \right) + \left( \pd{f}{r}{} \pd{r}{z}{} \cdot z + f \right)$$

where

$$\vec{r} = (x, y, z)$$

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no, it would just be the partial of r times f in each case

Like this?

$$\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } \nabla \cdot \vec{r}f(r) = \pd{(rf)}{x}{} + f\pd{(rf)}{y}{} + f\pd{(rf}{z}{}$$

or just 3f?

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My initial expression seems right to me now...

Is f a scalar field or a vector field?
Is it del.(r*f) or (del.r)(f)

I wrote it out just how they gave it to me but I assumed del.(rf) based on the question they ask after it.

f is scalar field, f(r), r = ||r||

There's a type of product rule for a scalar field times a vector field.