After testing with the exactly same stats (only variable is the artifact effect l) the Vile Ink does not give you 35% derivitive damage.
All artifacts are maxed, with the same 2 stats
(250 atk and 45% atk)
With +20 The Eye Ball of Giant (no passive effect) equipped, Ivellios passive does 9700 damage.
With the +20 Hapry Nail equipped (20% damage to passive), Ivellios passive does 10,400 damage.
With the +20 Vile Ink equipped, (35% damage to derivative damage) Ivellios passive does 10,100 damage.
The Vile ink does ‘something’ to the derivative damage bonus, but it should be 15% more than what the Harpy Nail outputs.