を元(もと)にBased on, From, According to

These phrases both mean 'based on' and can overlap, but differ slightly. にもとづいて is more formal and tends to show a concrete or logical foundation for an action or product. をもとに is less formal (but still used in formal contexts) and can indicate a general basis or inspiration where にもとづいて may not fit.

をもとに and にしたがって can both be translated as according to or based on, but differ in nuance. をもとに suggests a foundation or inspiration, as in 実話をもとに書いた小説 (a novel written based on actual events). にしたがって focuses more on being in line with or following something, as in 指示にしたがって (according to instructions). It can also mark proportional changes, as in 普及にしたがって (as it spread).

Both indicate a basis, but the type of basis differs. を前提に sets a condition or assumption that must hold true, often before action is taken, as in 合意があることを前提に話を進める ('We proceed with the discussion on the assumption that there is agreement.'). をもとに means ‘based on’ and refers to concrete materials, data, or sources actually used to create or decide something, as in 実話をもとに映画が作られた ('The movie was made based on a true story.').

Both involve a basis or source, but differ in usage. によると・によれば reports information, e.g. 天気予報によると雨が降るそうです ('According to the weather forecast, it will rain.') をもとに means doing something based on a source or foundation, e.g. 事実をもとに小説のストーリーを考えた ('I thought up a story for a novel based on true events.')

をもとに indicates a fixed source or basis for something, as in 実話をもとに小説を書く (Write a novel based on true events). に応じて shows flexible adjustment or response to conditions, as in 状況に応じて対応を決める (Decide the response according to the situation).