Here are the rules for damage taken while at 0 HP (Player’s Basic Rules Version 0.3, Page 76):
Damage at 0 Hit Points. If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage is from a critical hit, you suffer two failures instead. If the damage equals or exceeds your hit point maximum, you suffer instant death.
And on Page 75 are the following Instant Death rules:
Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.
For example, a cleric with a maximum of 12 hit points currently has 6 hit points. If she takes 18 damage from an attack, she is reduced to 0 hit points, but 12 damage remains. Because the remaining damage equals her hit point maximum, the cleric dies.
This means that even if you have more than 0 HP before you take the damage, such a hit is treated the same as the rule on Page 76 for damage taken while at 0 HP, at least as far as instant death is concerned.
So would this mean that if you take damage reducing you to 0, and there is still more damage remaining from that hit (but not enough to kill you outright), you would also immediately suffer a failed death saving throw? An example: you have 5 HP, and suffer a hit worth 6 damage, reducing you to 0 HP plus 1 remaining damage. Do you suffer the failed death saving throw in this instance, or do you only suffer the failure if you receive damage when you had 0 HP before any damage was dealt?
I researched this question but was not satisfied that this particular aspect of the rule was clarified there. That question contains an assumption that the scenario I’ve described does not result in a failed death saving throw (“I assume it won’t, like the first hit that brings you to 0 hit points”), but my interpretation of the rules doesn’t seem to come to that conclusion, or perhaps I’m misinterpreting?