Both were in some way derived, but the lineage is closer for the Democrats.
The Democratic-Republicans split into factions during the 1820s, resulting in the formation of the Jacksonian Democrats and the National Republicans, which became the Whigs. The National Republicans/Whigs were generally closer to the policies of the defunct Federalist party than to those of the Jeffersonian Republicans.
I think of the lineage as basically this:
Anti-Federalists --> Jefferson Republicans --> (major faction) Democratic-Republicans --> Jackson Democrats --> Democratic Party
Federalists --> ( faction within Democratic-Republican Party) --> National Republicans --> Whigs --> (+anti-slavery Democrats, minor factions) Republican Party
Don't forget that southern Whigs became Democrats after the Whig Party's collapse.