In the proof that the Schrodinger equation stay normalized as time goes on, it starts with moving the time derivative inside the integral, i.e. $$\frac{d}{dt}\int_{-\infty}^\infty |\Psi(x,t)|^2dx = \int_{-\infty}^\infty \frac{\partial}{\partial t}|\Psi(x,t)|^2 dx.$$
How is the swapping of the derivative and integral justified here? Is this mathematically justifiable or just something we assume holds in physics?