Creation/Annihilation operator help

[cahill8]

Homework Statement

I'm trying to go through something shown in a lecture and the part I'm stuck on is shown here.
$$\frac{\partial}{\partial t}\int^t_0 \mbox{du }e^{i \omega(t-u)}\hat{b}(u)=\hat{b}(t)$$

$$\hat{b}(u)$$ is an annihilation operator

The Attempt at a Solution

Can someone explain how this step is made? Obviously a delta function has to come in somewhere but I don't know how to do that with a derivative (only another integral).

 

[issacnewton]

this follows from a theorem in calculus

Theorem: Suppose that a function $g:[c,d]\rightarrow [a,b]$ (not necessarily a
onto function) is differentiable, and a function $f:[a,b]\rightarrow \mathcal{R}$
is continuous. If

$$H(t)=\int_a^{g(t)} f(u)\, du$$

with $t\in [c,d]$, then H is differentiable and

$$\large H'=(f(g(t))g'(t)$$

in your example

$$f(u)=e^{i \omega(t-u)}\hat{b}(u)$$

and $g(t)=t$

so

$$H'(t)=(f(g(t))g'(t)$$

$$g'(t)=1$$

so

$$H'(t)=f(t)$$

which gives what you are seeking