# Differentiation under the Integral Sign

Let $X$ be an open subset of $\mathbb{R}$, and $Y$ be a measure space.

Suppose that a function $f:X\times Y\rightarrow \mathbb{R}$ satisfies the following conditions:

1.$f(x,y)$ is a measurable function of $x$ and $y$ jointly, and is integrable over $y$, for almost all $x\in X$ held fixed.

2.For almost all $y\in Y, f(x,y)$ is an absolutely continuous function of $x$. (This guarantees that $\frac{\partial }{\partial x}f(x,y)$ exist almost everywhere.)

1. $\frac{\partial }{\partial x}f(x,y)$ is locally integrable. That is, for all compact intervals $[a,b]$ contained in $X$ $$\int_{a}^{b}\int_Y|\frac{\partial}{\partial x}f(x,y)|dydx<\infty$$

Then $\int_Yf(x,y)dy$ is absolutely continuous function of $X$, and for almost every $x\in X$, its derivative exists and is given by $$\frac{d}{dx}\int_Yf(x,y)dy=\int_Y\frac{\partial}{\partial x}f(x,y)dy$$