Differentiating
$\int_{(m-d-\mu)/\sigma}^{\infty}xf(x)dx$
with respect to $\sigma$, where $\sigma$ is the standard deviation of the standardarized random variable $x$ and $\mu$ its mean. I guess that it is
$\frac{m-d-\mu}{\sigma}f\left(\frac{m-d-\mu}{\sigma}\right).$
Correct me if I am wrong.