Derivative of parameter integral over $\mathbb{R}$

Question

Under which conditions on the integrand is it possible to differentiate under the integral
$$
\frac{d}{dx}\int_{-\infty}^\infty f(x,y)\, dy=\int_{-\infty}^\infty\frac{\partial}{\partial x}f(x,y)\, dy?
$$

I know Leibniz rule. But here, the integration over $\mathbb{R}$ irritates me and I dont know how to make sure that Leibniz Rule works.

Answer 1

If you want to differentiate at a point $x$ you need to verify :

1. For all $x$, $f(x,\cdot)$ is integrable
2. $f(\cdot,y)$ is differentiable at $x$ for almost every $y$.
3. There is an open interval $I$ containing $x$ and an integrable $g: I\times \mathbb{R} \to \mathbb{R_+}$ such that for all $z \in I$, $\left|\frac{df}{dx}(x,y)\right|\leq g(y)$ almost everywhere.