Good article, but I believe it lacks information what specifically these magical dFdx, dFdy, and fwidth = abs(dFdx) + abs(dFdy) functions are computing.

The following stackexchange answer addresses that question rather well: https://gamedev.stackexchange.com/a/130933/3355 As you see, dFdx and dFdx are not exactly derivatives, these are discrete screen-space approximations of these derivativities. Very cheap to compute due to the weird execution model of pixel shaders running in hardware GPUs.