The doctors did everything they could to save the dying old woman but failed.在文章中看到这个句子,我的疑惑是:这个句子中情态动词could后面跟了带to的不定式.有这样的用法吗?我只知道,情态动词后只跟动词原形.谁能解答下我的疑惑,to save...应该是不定式短语

这就要从你对这个句子的理解开始了，意思是什么。先看看。
the doctors did everything they could.
the doctors did everything(they could) to save the old man.

你说对了这里不定式做目的壮语从句，省略do是因为前面已经有了一个did,避免累赘，再给你举个例子：
every year it gets most difficult to separate web wheat from chaff and pick a handful of sites from billions
这句billions后面省略了of sites，因为前面有了这两个词，省略避免累赘。

To save the dying old woman, the doctors did everything they could, but failed.

这个不是不定式，是目的状语从句

everything（that） they could （do）省略了 that 和do 是定语从句
to save the dying old woman but failed.
目的状语

The doctors did everything they could to save the dying old woman but failed.
they could 是everything 的定语， to 后面的成分是做目的状语
医生做了他们所能做的一切来挽救这个垂死老妇人的生命，但失败了