I have a density matrix defined as

$\rho(t) = \int_{-\infty}^{+\infty} \sigma(t,x) \otimes |x\rangle\langle x|$

I want to show that 
$\hat{P} \rho(t) \hat{P} - \frac{1}{2} (\hat{P^2} \rho - \rho \hat{P^2}) = 2\frac{\partial^2 }{\partial x^2} \sigma(t,x)$ where $\hat{P}$ is the momentum operator. 

The question is similar to https://physics.stackexchange.com/questions/63614/density-operator-expectation-value-coherent-states, but this doesn't help me in getting the answer. $\hat{P} = -i \frac{\partial}{\partial x}$ in the position  basis, but I am unable to get the final answer. Can someone help me?