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There are a number of verbs where there is a 〜xasu → xaseru transformation to
produce an transitive verb from an intransitive, eg:

  * 死なす→死なせる
  * 生かす→生かせる
  * 飲ます→飲ませる

Is this some kind of generalized rule? (perhaps a classical conjugation?) Or
is it something that must be handled on a case-by-case basis?
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grammar verbs conjugations
Is there a general rule for deriving xasu→xaseru intransitives such as 死なせる from 死なす?
There is another base form before ~asu, and yes, there is rules for that.

```

    - 死ぬ (die)　　  ⇒　死なす　(let ~ die)  
                     ⇒　死なせる (make ~ die)
    - 生きる (live)   ⇒　生かす　(let ~ live)   
                     ⇒　生かせる (make ~ live)
    - 飲みます(drink) ⇒　飲ます　(let ~ drink) 
                     ⇒　飲ませる　(make ~ drink)
    
```
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2011-06-01T00:13:08.100
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Here are some pointers:

-Verbs ending in anything besides iru/eru can be called 五段, besides you conjugate them using the five rows in the kana table (one for each vowel). So use the five vowels for different contexts-

  * 死な- 死なない I won't die 死なせる To cause to die.
  * 死に- 死にますI'm gonna die now.
  * 死ぬ- 死ぬI'm gonna die/to die
  * 死ね- 死ね！ die! 死ね- 死ねば- if I die
  * 死の- 死のうとする to try to die

For most verbs that end in iru/eru, called 一段 because they conjugate on the
same vowel row, use these instead:

  * 食べない - I'm not gonna eat eat 食べさせる- To cause to eat
  * 食べます - I will eat
  * 食べる - I will eat
  * 食べろ - Eat!
  * 食べれば- If I eat
  * 食べようとする - to try to eat 

These forms are called 未然形、連用形、終止形、連体形、仮定形、命令形, repectively.

Don't forget, though: I say "most" because there are iru/eru verbs that are
五段. 走る, for example; the negative is 走らない.
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It seems that there are several confusions in the question.

First, 死なす and 死なせる are almost interchangeable (to make/let … die), and so are
飲ます and 飲ませる (to make/let … drink). (I wrote “almost” just in case; honestly
speaking, I cannot think of any situation where one is correct is the other is
not.)

Second, the relation between 生かす (to make/let … live; to utilize) and 生かせる
(can make … live; can utilize) is very different from the other two pairs of
words. The pair 生かす/生きさせる has the same relation as the two pairs 死なす/死なせる and
飲ます/飲ませる.

Both 飲ます and 飲ませる are made from the verb 飲む. 飲ます is a verb derived from the
verb 飲む (to drink). The derived verbs like 飲ます where the derivation adds the
meaning “make/let” are called 使役動詞 in the Japanese grammar. 飲ませる consists of
飲ま (the 未然形 of 飲む) and an auxiliary verb せる (make/let). The same applies to
死なす and 死なせる, both of which are made from the verb 死ぬ (to die).

Almost the same explanation applies to 生かす and 生きさせる, both of which are made
from the verb 生きる (to live), but there are two complications here.

  * The verb 生かす has an additional meaning: to utilize. Therefore, 生かす and 生きさせる are not always interchangeable.
  * The phrase 生きさせる consists of 生き and させる. 生き is the 未然形 of 生きる, and させる is an auxiliary verb which has the same meaning as せる. The distinction between せる and させる depends on the verb to which it is attached, but I will not go into details now.

生かせる is a verb derived from the verb 生かす using another method of derivation,
and it means “can make … live” or “can utilize / can be utilized.”

_Edit_ : In revision 1, I used the term “conjugation” incorrectly.
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All three cases are actually different.

> 死なす→死なせる

In the standard (Tokyo) dialect, `死なせる` is the correct form, and `死なす` is
marginal. If `死なす` is allowed in any case, then `死なせる` will not be allowed,
and hence these should be considered different dialects/idiolects used with
the same meaning. It is the causative form of `死ぬ` (See below).

> 生かす→生かせる

In the standard dialect, `生かす` is the correct form, and `生かせる` is completely
ungrammatical (unless you are talking about the potential form).

> 飲ます→飲ませる

Here, you see the **causative** morpheme `-(s)ase-`. When the verb stem ends
with a consonant, the form `-ase-` is used.

> kak-ase-ru 'cause someone to write something'
>
> kag-ase-ru 'cause someone to smell something'
>
> kas-ase-ru 'cause someone to lend something to someone'
>
> kat-ase-ru 'cause someone to win'
>
> kam-ase-ru 'cause someone to bite something'
>
> sin-ase-ru 'cause someone to die'
>
> nom-ase-ru 'cause someone to drink'

When the verb stem ends with a vowel (`i` or `e`), the form `-sase-` is used.

> mi-sase-ru 'cause someone to see'
>
> tabe-sase-ru 'cause someone to eat something'
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The basic answer:「せる」is the newer form and 「す」is classical. Both are correct
and they have the same meaning.

But, as @user458 points out, the newer form is stronger in the Standard
Japanese and the classical form is stronger in the Kansai dialect.

Also, as @Tsuyoshi Ito points out, some verbs have developed different
meanings for the two forms, even though historically they were the same.

Also, there are other, subtler reasons you prefer one over the other depending
on the verb.

The latter two are the reasons why the classical form is still sometimes used
in the Standard Japanese.
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