延期する

That's a great question! Both 延期する (enki suru) and 先送りする (sakurasu suru) mean 'to postpone' or 'to defer,' but they have slightly different nuances. 延期する is generally used for more formal or official postponements, like events, meetings, or deadlines. It often implies a new date or time will be set. For example:
例: 会議は来週に延期された。(The meeting was postponed to next week.)

先送りする is a bit more general and can be used for less formal situations, or when you're simply putting something off without necessarily setting a new date. It can sometimes carry a nuance of procrastination. For example:
例: 宿題を先送りにしてしまった。(I put off my homework.)

Yes, you can definitely use 延期する (enki suru) for a doctor's appointment! It's perfectly natural in that context.
例: 歯医者の予約を来月に延期しました。(I postponed my dentist appointment to next month.)

While 延期する (enki suru) itself is not impolite, when you're asking someone to postpone something, you might use more polite phrasing around it. For instance, you could say:
例: 恐れ入りますが、会議を延期していただけませんか？ (Excuse me, but would you mind postponing the meeting?)
Or if you're stating that something has been postponed, you can use humble or respectful language depending on the situation.

When you're the one postponing something, you can simply use 延期する (enki suru) with the direct object.
例: 私は旅行を延期しました。(I postponed the trip.)
例: 試験は延期されました。(The exam was postponed.) (Here, it's a passive construction, implying someone else postponed it or it was decided.)

Good question! Let's break down the kanji:
延 (en): This kanji means 'extend,' 'prolong,' or 'delay.'
期 (ki): This kanji means 'period,' 'term,' or 'date.'
So, together, 延期 (enki) literally means to 'extend the period/date,' which leads to its meaning of 'postpone' or 'defer.'

While 延期する (enki suru) typically implies a postponement for a longer period (days, weeks, months), you *could* technically use it for a few hours, but it might sound a bit formal for such a short delay. For a very short delay, you might hear things like:
例: 少し遅れます。(I'll be a little late.)
例: 少し時間をずらしましょう。(Let's shift the time a little.)

The plain form of 延期する (enki suru) is 延期する itself for the non-past affirmative.
Plain past: 延期した (enki shita)
Plain negative: 延期しない (enki shinai)
Plain negative past: 延期しなかった (enki shinakatta)

延期する (enki suru) is common in both spoken and written Japanese, especially when discussing formal or semi-formal postponements of events, meetings, or plans. You'll hear it in daily conversation and see it in news articles or official announcements.

Yes, there are a few common patterns. You often hear:
〜が延期される (ga enki sareru): ~ is postponed (passive voice)
例: 試合が雨で延期された。(The game was postponed due to rain.)
〜を延期する (o enki suru): to postpone ~ (active voice)
例: 私たちはイベントを延期しました。(We postponed the event.)

To say 'indefinitely postponed' using 延期する (enki suru), you would add words like 無期限に (mukigen ni) meaning 'indefinitely' or 未定 (mitei) meaning 'undecided/unfixed.'
例: 会議は無期限に延期された。(The meeting was indefinitely postponed.)
例: 開催日は未定で延期されました。(The holding date is undecided, and it has been postponed.)