Question about the derive of filter roll-off
For any arbitrary rational complex function $H(\omega) = \frac{N(\omega)}{D(\omega)}$,

$$
|H(\omega)| = \frac{|N(\omega)|}{|D(\omega)|}
$$

This follows straightforwardly from the properties of the polar (i.e. magnitude/phase) representation of complex numbers. So, for your example:

$$
|H(\omega)| = \frac{|1|}{|1 + j \frac{\omega}{\omega_c}|}
$$

The magnitude of an arbitrary complex number is:

$$
|a + jb| = \sqrt{a^2+b^2}
$$

So for your example:

$$
\begin{align}
|H(\omega)| &amp;= \frac{1}{\sqrt{1^2 + \left(\frac{\omega}{\omega_c}\right)^2}} \\
&amp;= \frac{1}{\sqrt{1 + \left(\frac{\omega}{\omega_c}\right)^2}}
\end{align}
$$