The rules say you provoke an opportunity attack when you leave someone's range, But how does that work if you're dual wielding a whip and a dagger since you have two different attack ranges?

I would think it would depend on what is in your main-hand. The off-hand weapon can only be used for the bonus action so I would say an opportunity attack always uses whatever weapon is in your main-hand.

There isn't a "main hand" in 5e.

It is either you can make an AoO with the weapon when the target leaves either weapon's range, or you can make an AoO with either weapon when the target leaves the further range.

I'd go with the first option (you can use the dagger if they move out of 5 feet and the whip if they move out of 10 feet). You only have 1 reaction, so it isn't over powered.

I would rule as emmote said after seeing that the reach is a possibility from https://www.sageadvice.eu/2018/03/15/if-i-have-a-reach-weapon-do-adjacent-enemies-provoke-attack-of-opportunity/

If you are mainhanding a reach weapon, I would think that they would have to leave that 10 ft area, meaning that they could skirt around you to attack your allies, barring something like sentinel if it has a non shield carrying application. If someone said that they were ambidextrous, I'd probably ask which hand they would carry a shield in and rule that the other is their dominate/main hand.

I'm flexible with the idea, while I agree that most people are restricted by "handedness" the rules are pretty ambiguous about that. When looking at Two Weapon Fighting it states:

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.

Emphasis is mine, but it clearly does not designate main or off hand simply that the weapons are held in different hands. So, that being the case, I would leave it up to the player to choose when to use which weapon. They could use the Dagger as soon as the enemy leaves the range of that attack or they could wait and use the Whip when the enemy leaves that attack range. There is no reason to limit which weapon they choose to use, but they are only allowed to use one of those weapons as a Reaction (assuming there are no exceptions created by Feats or Abilities).

There is no "main hand" in 5e.

You can attack with right hand as action and left hand as bonus action one turn, then attack left hand with action and right hand with bonus the next turn.

[Edit]Sniped by DMThac0 (who seems to agree with my opinion)

Yea after looking at the AoO rule and Two-Weapon Fighting I'm retracting my first reply. I think choosing which to trigger would be fine.