From the spreadsheet that Kotono linked, it looks like each level of a nymph costs (1+sqrt(5))/2 (a.k.a., the golden ratio) times as much as the previous level. All of the costs displayed are pretty close to that and probably within rounding of it. It rounds or truncates the numbers that are displayed on the screen, so it's likely that the exact value is either computed as a base value times the golden ratio to whatever power, at least up to rounding.

Damage dealt per level of a nymph also looks like it scales at the same ratio. Thus, the total cost to level a nymph to 10 is about 197 times the cost of unlocking the nymph in the first place. The DPS (excluding super nymph skills) that the nymph will deal at level 10 is about 76 times what it will deal at level 1. That doesn't give you an easy way to predict super nymph skills, however.