Critical hit means, for each damage die you would roll for a non-critical, you roll it twice instead of once.
Death strike means you take the damage you would have dealt if it were not a death strike (which would still be a critical hit), and you double that.
The order of operations is:
- Decide which damage dice, and how many of them, to roll.
- Roll them and add them up.
- Add any relevant damage modifier, e.g. your dexterity modifier or the +10 modifier from the Sharpshooter feat.
- Multiply that sum by any relevant multiplier, e.g. multiply by 2 for Death Strike.
Therefore:
- Normal rapier damage = 1d8 + dex mod
- Rapier damage + sneak attack = 1d8 + 10d6 + dex mod
- Critical (rapier damage + sneak attack) = 2d8 + 20d6 + dex mod
- Death strike (critical (rapier damage + sneak attack)) = 2 * (2d8 + 20d6 + dex mod)
Assassin is a highly specialized class archetype, and it is a huge investment to put 17 levels into it that you could have put into other classes. The reward for that investment is that the assassin is unmatched in the amount of damage it can inflict on a surprised target. If you ask me, an average of 168 damage (on a failed save) is the least the game could do to reward that investment.
According to the lead rules designer, it seems the answer is that they are different sources.
The general rules on how to apply damage from an attack are a little vague in this situation:
3. Resolve the attack. You make the attack roll. On a hit, you roll damage, unless the particular attack has rules that specify otherwise. Some attacks cause special effects in addition to or instead of damage.
The Wyvern does have special attack rules and, as Dale M has pointed out there is an extra conditional "gate" on the second poison damage which would pertain to the "unless" clause of the rule above (while the first half of the damage does not).
It is also illustrative to note how the DMG talks about poisons. Wyvern Poison is an injury poison (meaning it is applied in conjunction with a weapon attack, not inhaled, ingested, or applied topically):
Injury. A creature that takes slashing or piercing damage from a weapon or piece of ammunition coated with injury poison is exposed to its effects.
This description of how the poison works talks about its damage as separate from the weapon's damage which indicates (to me at least) that it is intended to be treated as a separate source.
In an unofficial tweet on a similar subject, lead rules designer Jeremy Crawford discussed the process of resolving the save on a Solar's Slaying Longbow which also features a two-part damage/effect combination:
Slaying Longbow. Ranged Weapon Attack: +13 to hit, range 150/600 ft., one target. Hit: 15 (2d8 + 6) piercing damage plus 27 (6d8) radiant damage. If the target is a creature that has 100 hit points or fewer, it must succeed on a DC 15 Constitution saving throw or die.
Crawford explains the way to handle the extra effect rider on the bow attack thus:
Slaying Longbow sequence:
Hit
Deal damage (apply any resistance)
Make saving throw if the target has hit points ≤ 100
Extrapolating this interpretation to the Wyvern or other compound damage attacks requiring a save: if the target is reduced to zero hit points by the first set of damage and then fails the save, the additional damage would cause a failed death saving throw because it happens after the first set of damage is calculated and applied as a result of the poison getting into the wound created by the weapon (or claw/fang, whatever).
However, I think it would also be a reasonable house rule to have the player make the Con save before damage was announced and then present all the damage at once to obviate the failed death save.
Best Answer
No, the poison's damage is not part of the attack and therefore not doubled
Death strike explictly states that the attack is doubled (emphasis mine):
Damage from poison applied to a weapon has been clarified as being separate damage from the attack entirely, due to needing a saving throw, as you have referenced in your question. We can phrase this in this manner due to the ruling that poison damage is not doubled on a critical hit, and the definition of critical hit damage (emphasis mine):
A couple clarifications here from the bolded text.