I was just wondering...when a paladin divine smites, is that part of the damage DC counted for rolling to see if you maintain a spell?
Divine Smite adds to weapon damage for Concentration saves
The description of the Divine Smite feature (p. 85 PHB) says:
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one paladin spell slot to deal radiant damage to the target, in addition to the weapon’s damage.
The damage is added to the weapon's damage on the same attack. Both together make up the damage from the attack, and that is the basis for the Constitution save to maintain concentration.
The Concentration section on PHB p. 203 defines how taking damage leads to a Constitution save:
Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
It does not care about what kind of damage, or if it is part of the weapon damage or not, but when you take the damage. All the damage is taken from the same attack and at the same time, not from multiple sources, so Divine Smite will not trigger two separate saving throws.