Differentiating the limits of a definite integral - multivariable

Question

How would I go about finding the following: $$ \frac{\partial^2}{\partial x \partial y} \left( \int_x^{x-2y}\frac{1}{\sqrt{2 \pi t}} e^{-z^2/2t} dz \right) $$

Answer 1

\begin{align*} \frac{\partial^2}{\partial x \partial y} \left( \int_x^{x-2y}\frac{1}{\sqrt{2 \pi t}} e^{-z^2/2t} dz \right) & = \frac{\partial}{\partial x} \left( \frac{\partial }{\partial y} \left( \int_x^{x-2y}\frac{1}{\sqrt{2 \pi t}} e^{-z^2/2t} dz \right) \right) \\ & = \frac{\partial}{\partial x} \left( \left[ \frac{1}{\sqrt{2 \pi t}} e^{-(x-2y)^2/2t} \right] \frac{\partial}{\partial y} (x-2y) + \left[ \frac{1}{\sqrt{2 \pi t}} e^{-(x)^2/2t} \right] \frac{\partial}{\partial y} (x) \right) \\ & = \frac{\partial}{\partial x} \left( \frac{-2}{\sqrt{2 \pi t}} e^{-(x-2y)^2/2t}\right) \\ & = \frac{\sqrt{2} e^{-(x-2y)^2/2t}(x-2y)}{t\sqrt{\pi t}} \end{align*}