Find the inner product of the Pauli matrices and the momentum operator?

Homo Novus

1. The problem statement, all variables and given/known data

Show that the inner product of the Pauli matrices, σ, and the momentum operator, [itex]\vec{p}[/itex], is given by:

σ [itex]\cdot[/itex] [itex]\vec{p}[/itex] = [itex]\frac{1}{r^{2}}[/itex] (σ [itex]\cdot[/itex] [itex]\vec{r}[/itex] )([itex]\frac{\hbar}{i}[/itex] r [itex]\frac{\partial}{\partial r}[/itex] + iσ [itex]\cdot[/itex] [itex]\vec{L}[/itex]),

where [itex]\vec{L}[/itex] is the angular momentum operator and [itex]\vec{r}[/itex] is the displacement vector.

2. Relevant equations

p[itex]_{x}[/itex] = [itex]\frac{\hbar}{i}[/itex] [itex]\frac{\partial}{\partial x}[/itex]
[itex]\vec{L}[/itex] = [itex]\vec{r}[/itex] × [itex]\vec{p}[/itex]

3. The attempt at a solution

I figured that I could write:

[itex]\vec{p}[/itex] = [itex]\frac{\hbar}{i}[/itex] [itex]\frac{\partial}{\partial r}[/itex] [itex]\hat{r}[/itex]

So then:
σ [itex]\cdot[/itex] [itex]\vec{p}[/itex] = (σ [itex]\cdot[/itex] [itex]\hat{r}[/itex]) [itex]\frac{\hbar}{i}[/itex] [itex]\frac{\partial}{\partial r}[/itex]
= [itex]\frac{1}{r}[/itex] (σ [itex]\cdot[/itex] [itex]\vec{r}[/itex]) [itex]\frac{\partial}{\partial r}[/itex]

... But that clearly gets me nowhere. Help?