Strength of the grave question concerning the "damage taken"

In the PHB, "Strength of the Grave" allows you to make a DC 5+ damage taken to drop to 1 HP instead of zero hit points. Based on what the PHB says, I would like to clarify it before my next session. For Example, in 5E if you have 5 health remaining but an attack against you deals 10 damage, you technically only take 5 damage as you drop to zero (because there is nothing past zero).

So in this example would you roll on a DC 5+10 or DC 5+5? I am right in assuming it would only be DC =5 + 5(damage taken). Does this mean that sometimes damage taken does not equal damage dealt?

BUUUUT... you can also straight-up die if the excess damage equals your total HP (no saving throws you're just dead). 

I'm wanting to clarify this so i can have a better understanding of how the rules are meant to be applied to this sub-class feature.

Thanks in Advance! :)

In your example it is DC 5+10 so DC 15. The damage taken was still 10. You just don’t go into negative numbers. 

DC 15. The damage taken is damage dealt, not the amount of hit points your total was reduced by.

This roll is pretty difficult, making this class feature a painful and futile exercise a lot of the time. Once you get any distance into your progression you'll be taking damage numbers that make this saving throw very unlikely to succeed.

In my own game I house rule this to use the same formula as a concentration check. Charisma check at DC 10 or half the damage taken, whichever is higher. It simply allows the class ability to have a reasonable chance of activating. Honestly I don't think this class ability would be overpowered if it could be activated with no check at all (like the Half-Orc racial ability Relentless Endurance), but giving it an easier DC at least splits the difference.

Don't forget you can't activate Strength of the Grave against a crit, or against radiant damage, so even if it required no roll to activate it would still not be as good as Relentless Endurance.

Yes - great idea.

Damage taken and hit points lost are separate in this case. You took 10 damage, but you only lost 5 hit points.

I think I would house-rule this one to work like the Barbarian's Relentless Rage feature. Start at DC5, then each time you use it before a Long Rest you increase the DC by 5 so it is easy to use it the first time but gets harder to make the save as you use it more times before getting a rest.