I understand the following equation to be correct, but why can we treat the differentials as fractions and cancel them out? What would be the correct way to view it?
$$ \int_{-\pi/a} ^{\pi/a}dk_xf(k_x)\frac{d^2}{dk_x^2}[\delta(k_x)] = \int_{-\pi/a} ^{\pi/a}f(k_x)d \left(\frac{d}{dk_x} [\delta(k_x)] \right) $$