に応えて details

にこたえて

in response to; living up to N (an expectation, request, or hope)

彼かれはファンの声援せいえんに応こたえて、アンコールで一曲いっきょく歌うたった。

kare wa fan no seien ni kotaete, ankooru de ikkyoku utatta.

In response to the fans' cheers, he sang one more song as an encore. neutral

みんなの期待きたいに応こたえたくて、めちゃくちゃ練習れんしゅうしたんだ。

minna no kitai ni kotaetakute, mechakucha renshuu shita n da.

I wanted to live up to everyone's expectations, so I practiced like crazy. casual

Examples

in response to; living up to — an action satisfies an expectation, hope, or request

彼かれはファンの声援せいえんに応こたえて、アンコールで一曲いっきょく歌うたった。

kare wa fan no seien ni kotaete, ankooru de ikkyoku utatta.

In response to the fans' cheers, he sang one more song as an encore. neutral

みんなの期待きたいに応こたえたくて、めちゃくちゃ練習れんしゅうしたんだ。

minna no kitai ni kotaetakute, mechakucha renshuu shita n da.

I wanted to live up to everyone's expectations, so I practiced like crazy. casual

当社とうしゃは、社会しゃかいの要請ようせいに応こたえて、新あたらしい環境かんきょう方針ほうしんを策定さくていいたしました。

tousha wa, shakai no yousei ni kotaete, atarashii kankyou houshin o sakutei itashimashita.

In response to demands from society, our company has drawn up a new environmental policy. formal

に応え — て-drop for written narration and reports — 声援せいえんに応こたえ、力走りきそうした; not used in conversation.
に応えた — modifies a following noun directly — 期待きたいに応こたえた活躍かつやく "a performance that lived up to expectations"; に応えて cannot sit in that slot.

に対して
Both put a noun before a て-form action and both can describe reacting to something, but に応えて specifically requires the noun be an expectation, request, or hope that gets FULFILLED by the action (期待に応えて頑張った — a live-up-to relation), while に対して is directionally neutral and works equally for negative or adversarial responses (苦情に対して謝罪した — a complaint that is being addressed, not satisfied) — に応えて cannot substitute there since nothing is being lived up to.

に応じて
One character apart and built on the same 応 kanji, and both genuinely fit a request — but 応じる frames the reaction as COMPLYING with or tracking a condition (要求に応じて値下げした "lowered the price in compliance with the demand"), while 応える frames it as LIVING UP TO someone's hope (要求に応えて値下げした "lowered the price to satisfy what they wanted") — swap them and the same sentence shifts from grudging compliance to a generous response.