[KyleGoetz]

Taken from the same book as the other lesson I've posted. Note that these four "mini-lessons" are for the former JLPT1 and 2, so don't be sad if it's something you haven't encountered. The first two are former JLPT2, and the latter two are JLPT1. ために is likely missing because it's so basic compared to these.

Again, translated by me.

Sections:
〜ように
〜上で
〜べく
〜んがため（に）

===ように===
meaning: an intention of doing something in order to make something a reality
~ is a verb that does not express something within the controllability of the speaker
the verb is in dictionary or nai form

Examples:
かぜが早く治るように注射を打ってもらいました。
In order to quickly get over this cold, I received an injection.

「黒板の字がよく見えるように前の席に座りましょう。 」
"Let's sit in a front seat so that we can easily see the writing on the blackboard."

===上で===
Expresses an important goal.
Preceding 上で must be either dictionary form, or the pre-する form of a する verb plus の
e.g., 生活の上で comes from 生活する as you'll see in an example below.

Examples:
テレビは外国語の勉強の上でかなり役に立ちます。
TV is very useful for studying a foreign language.

数学を学習する上で大切なことは、基礎的な事項をしっ かり身につけることだ。
An important thing for studying mathematics is mastery of fundamental things.

Note: What comes after 上で must be an important or necessary thing.
For example, it is incorect to use in the sentence "For Japanese lifestyle, I bought many things."
But it is acceptable if you say "I bought necessary things" or "what are necessary things to buy?"
日本の生活の上で、必要なものは何ですか。

===べく===
Used when you want to say that you had some goal and made it so.
It cannot be followed by a command or request.
Preceded by dictionary form of a verb.

Examples:
インドのラムさんを迎えるべく空港まで行ったが、会え なかった。
To meet Lum from India, I went to the airport, but I couldn't find her.

彼女は新しい気持ちで再出発するべく、長野県の山村に 引っ越して行った。
In order to get a fresh start with new emotions, the girl moved to a mountain village in Nagano-ken.

===んがために===
When you want to say "I definitely wanted to make real in an affirmative way some goal, and I did something in furtherance of the goal."
Preceded by the nai form of the verb after you remove the nai, exception: する＞せ
Like with べく, it cannot be followed by a command or request.

Examples:
研究を完成させんがため、彼は昼夜寝ずに頑張った。
In order to complete his research, he worked hard without any sleep, day or night.

これも勝たんがための練習だから、がんばるしかない。
This, too, is practice/training for the purpose of winning, so you've just got to work hard.