I'm assuming $\frac{d}{dw}$ should be written as $\frac{\partial}{\partial \omega}$
I'm a bit confused by the part highlighted in green. I'm think i'm right in saying that when we integrate wrt one variable ($x$ in this case) we leave the other variable ($\omega$) constant. However the gren underline implies we that we can treat $\omega$ as a variable inside the integral which is integrating wrt $x$?
Think about an easier example. You want to rewrite $\int_0^1 2xy dx$ as $\frac{df}{dy}$ for some function $f$. (I realize this example is artificial, please bear with me.)
You notice that $2xy=\frac{\partial}{\partial y} xy^2$, so you have
$$\int_0^1 2xy dx = \int_0^1 \frac{\partial}{\partial y} xy^2 dx.$$
Now you justify taking the derivative out of the integral to get
$$\int_0^1 2xy dx = \frac{d}{dy} \int_0^1 xy^2 dx.$$
At no point here is $y$ actually changing inside the integration, we're just writing one function of $y$ as the derivative of another function of $y$.
The same happens in your example: you're rewriting $f(x,\omega)=xe^{-i \omega x}$ as $\frac{\partial}{\partial \omega} \left ( i e^{-i \omega x} \right )$. The value of $\omega$ isn't actually changing.