Leibniz integral rule

Statement

Suppose $f$ is a function of two variables $x,y$ and $f$ and its partial derivative $\partial f/\partial x$ are both continuous functions on an interval of the form $[x_1,x_2]\times [y_1,y_2]$. Then, for $x\in (x_1,x_2)$, we have:

$\frac{d}{dx}\underset{y_1}{\overset{y_2}{\int }}f(x,y)dy=\underset{y_1}{\overset{y_2}{\int }}\frac{\partial }{\partial x}f(x,y)dy$