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.
This is the Leibniz integral rule.
$$\frac{\partial}{\partial t} \int_{a(t)}^{b(t)} \; f(x,t,s) \,\mathrm{d}s = f(x,t,b(t)) \frac{\mathrm{d}}{\mathrm{d}t}b(t) - f(x,t,a(t))\frac{\mathrm{d}}{\mathrm{d}t}a(t) + \int_{a(t)}^{b(t)} \; \frac{\partial}{\partial t} f(x,t,s) \,\mathrm{d}s $$
Since your $a(t) = 0$, the second term on the right vanishes. Application of this expression has conditions: $f(x,t,s)$ and $\frac{\partial{f(x,t,s)}}{\partial t}$ is continuous in both $t$ and $s$ in an open neighborhood of $(x,t,s) \in \{x\} \times \{t\} \times [a(t),b(t)]$.