differentiation boundary of integral

Question

I got the following problem

$\int_{\ |x|\le t +1}f(x,t)dx=E(t)$

That might sound like a stupid question, but how would I got about finding $E'(t)$ ?

Would appreciate any help

Answer 1

Let $E(t)$ be given by

$$E(t)=\int_{-(t+1)}^{t+1}f(x,t)\,dx$$

Then, assuming the both $f(x,t)$ and $\frac{\partial f(x,t)}{\partial t}$ are continuous, the derivative, $E'(t)$, of $E(t)$ is given by Leibniz's Rule

$$E'(t)=f(t+1,t)+f(-(t+1),t)+\int_{-(t+1)}^{t+1}\frac{\partial f(x,t)}{\partial t}\,dx$$