For uni I'm working on Brownian Motion and need to derive
$$ -\frac{\partial^2}{\partial x \partial y} \frac{1}{\sqrt{2\pi t}}\int_{2y-x}^{\infty}e^{-u^2/(2t)}du, $$
into $$ \sqrt{\frac{2}{\pi}} \frac{(2y-x)}{t^{3/2}}e^{-(2y-x)^2/2t}\mathbb{I}_{\{x < y\}}. $$ The problem with this expression is that there is not a closed form integral of the gaussian curve, so just filling in and deriving will not work. How do I work around this? Thanks in advance!!