Lets assume we define some function 

$u(x,t):= \int\limits\_0^t u(x,t;s) ds$

Then there is a part in my lecture where it says

$\dfrac{\partial}{\partial t} u(x,t)= u(x,t;t)+ \int\limits\_0^t \dfrac{\partial}{\partial t} u(x,t;s) ds$

I don't understand where this comes from. 

The topic here is to solve the inhomogenous wave equation using Duhamel principle, but I think that the part I don't understand here has nothing to do with the equation but is just a simple integration rule I don't see.