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:

${\frac {d}{dx}}\int _{y_{1}}^{y_{2}}f(x,y)\,dy=\int _{y_{1}}^{y_{2}}{\frac {\partial }{\partial x}}f(x,y)\,dy$