Suppose $F(x;\mu,\sigma^2,0,1)$ is the CDF of a $N(\mu,\sigma^2)$ random variable $x$ truncated on the unit interval $(0,1)$. I'd like to show that $\frac{\partial}{\partial \mu} F(x;\mu,\sigma^2,0,1) < 0$ for any specific $x \in (0,1)$. I can empirically see this to be true but have difficulty analytically prove it.
To see an analogy, consider the (un-truncated) normal CDF $\Phi(\frac{x-\mu}{\sigma})$. It is well known that $\frac{\partial}{\partial \mu} \Phi(\frac{x-\mu}{\sigma})=-\phi(\frac{x-\mu}{\sigma})/\sigma <0$ for any $x\in R$, as $\sigma>0$ and $\phi(x)>0$ for all $x$. I was trying to extend the result in the un-truncated normal to truncated normal on the unit interval.