$ \frac{1}{f^2} $ Weighting for Least Squares FIR
Well the page you link to says:


  'differentiator' for type III and type IV filters, using a special weighting technique. For nonzero amplitude bands, the integrated squared error has a weight of (1/f)2 so that the error at low frequencies is much smaller than at high frequencies. For FIR differentiators, which have an amplitude characteristic proportional to frequency, the filters minimize the relative integrated squared error (the integral of the square of the ratio of the error to the desired amplitude).


Bolding mine.

So, following the paper:
$$
\epsilon_2 = \int_0^\pi W(\omega)(A(\omega)−D(\omega))^2d\omega
$$
for the differentiator, we want:
$$
W(\omega) = \frac{1}{\omega^2}\\
D(\omega) = \omega
$$
and $A(\omega)$ has the same form as before.

So
$$
\epsilon_2 = \int_0^\pi \frac{1}{\omega^2}(A(\omega) - \omega)^2 d\omega\\
= \int_0^\pi \left[\frac{A^2(\omega)}{\omega^2} - 2\frac{A(\omega)}{\omega} + 1 \right] d\omega
$$
Just substitute the expression for $A(\omega)$ and you're done.