The pills Derpy's mother takes to stop crying are magnesium pills.
Some women cry due to PMS, and magnesium can prevent that. I'm in no way saying that she's abusive due to PMS, it's just a possible explanation for the pills.Alternatively, the pills are a recreational drug.
Part of Derpy's mother realises that what she's doing is wrong, so she cries about it, but then she takes drugs to take her mind off it.Derpy's disability is unique to the MLP universe.
Which is why it's so vaguely defined.