Leibniz integral rule

The Leibniz integral rule, (The Leibniz rule of integration), of differentiation under the integral sign is the statement that

$\dfrac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x,b(x))b'(x) - f(x,a(x))a'(x) + \int_{a(x)}^{b(x)} \dfrac{\partial}{\partial x} f(x,t) dt$

In the case that $a(x)=a,b(x)=b$ are constants, then

$\dfrac{d}{dx} \int_{a}^{b} f(x,t) dt = \int_{a}^{b}\dfrac{\partial}{\partial x} f(x,t) dt$

This allows for the integral and differential operators to be interchanged.