Differentiating a definite integral

[gaganaut]

I was reading on a textbook and came across a step and I had no idea how the author got the next step from it. This was basically differentiation of a definite integral.

So here it is.

$$\frac{\partial}{\partial t}{\int_0^t}{\int_0^1}u_t^2(x,s)dx ds$$

So I figured that $$u_t=\frac{\partial u}{\partial t}$$. But isn't there a simple rule to differentiate an integral like this. I can probably get away with just using the final result from the book in my research, but not knowing how I got to it is just something that is bugging me.

So any help will be much appreciated.

Thanks

 

[JJacquelin]

What is the meaning of u_t ?
You write u(x,s). Is it a three argument function u(x,s,t) ?

 

[Wizlem]

Barring that u depends on t, the fundamental theorem of calculus will just drop the outer integral and replace s with t. Otherwise I think its valid to apply the chain rule in this fashion
$$\frac{\partial}{\partial t}{\int_0^t}{\int_0^1}u_t^2(x,s)dx ds = }{\int_0^1}u_t^2(x,t)dx + {\int_0^t}{\int_0^1}\frac{\partial}{\partial t}u_t^2(x,s)dx ds$$