Math Div expression help

Is the following true?

[tex]
\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)[/tex]

where

[tex]\vec{r} = (x, y, z)[/tex]

 

Like this?

[tex]
\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}{}
[/tex]

or just 3f?

 

My initial expression seems right to me now...

 

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||