If $f_0^0$ is the Fermi-Dirac distribution function, and if the first line of the attached Fig is correct, then is the second line correct? ∂_{a/b} are derivative over $k_x$ and $k_y$. If the answer is "no", how one can calculate the integral of the second line?

$\int= \int dk_x dk_y$

Many thanks in advance