He didn't know why or how she would die, only that she would. So he is pretty desperate and jumps to any chance he has, even turning to the dark side.

In-universe:

Anakin's need to control everything and inability to cope with uncertainty essentially drove him mad with paranoia.

PADME: you're not all-powerful ...

ANAKIN: Well, I should be! Someday I will be. I will be the most powerful Jedi ever. I promise you. I will even learn to stop people from dying! _{- source}

Not wanting to experience the pain he felt at his mother's death again, Anakin is desperate for anything resembling a solution.

_{Out-of-universe:
Lucas probably didn't think of that.}

Anakin is self-confident to a fault

The arrogance and pride that Anakin had - that Palpatine encouraged - was nearly limitless. He was about 22 years old and never planned ahead. He would wait until the last second to perform a task, wouldn't plan it out, and in his experience and according to his perspective everything always worked out well. He would even often ignore the advice or planning of his master in favor of his own last-minute decisions, and felt justified in doing so when things worked out.

It was this hubris that led him to take no action in the months, weeks, and days leading up to the birth event which he had foreseen was fatal to Padme.

The only thing he truly ever felt failure in was saving his mother's life, and he may have believed that if he had been there it would have been ok. This doesn't lead him to prepare in any special way for the impending birth, but to stick close to Padme, reluctant to leave her side without assurances from Palpatine that she would be watched over.

Anakin honestly believed that as long as he is near Padme, then no matter what happened he would be able to save her.

He didn't consider it a specialty, a process, or something he needed to prepare for. He assumed that he simply had enough power, and that he would learn anything additional he lacked on the fly during the event.

Perhaps Anakin could have saved her

We don't know what caused her death, not in terms that would help medical bots save her, anyway. However, there is evidence that as Darth Vader, Anakin had to use the force to maintain his own life and ability to move.

In a parallel universe, perhaps Anakin noticed her body was preparing for labor, and perhaps he stopped fighting with her and went to the medical facility, and perhaps he spent the rest of his life using the force to sustain her life. This one different choice would have him spending his force ability sustaining as much life in his wife as he did spend sustaining his own life as Vader.

This is, perhaps, something even the Jedi could have done, except for their creed preventing attachments. In fact, this creed points to the idea that if a Jedi used their power this way, where would they draw a line, assuming it's finite?

Darth Sidious's master, Darth Plagueis conducted all sorts of weird and wonderful experiments to create life and maintain life. These are detailed in Darth Plagueis novel which is prequel to Episode 1.

Sidious himself tells Annie this shortly before he reveals himself in Episode III;

"Darth Plagueis was a Dark Lord of the Sith so powerful and so wise, he could use the Force to influence the midi-chlorians to create life. He had such a knowledge of the dark side, he could even keep the ones he cared about from dying."

-Palpatine

Old Sid thought that by teaming up with young Annie they would be able to use their combined mastery of the dark side and influence over the midi-chlorians to complete the job Plagueis could not, to be master of over life and death itself...including the ability to keep the ones you love from dying.

Young Annie had visions of his wife dying, having bit of a history with death vision coming true, he rightful concluded that his visions would come to pass into reality.

So having the power of stopping people you love, e.g. your wife from shuffling of this mortal coil before their time, is something I would suggest is a pretty useful skill to have in ye-old repertoire.

Other answers mention that Anakin is arrogant and not that smart, which it true, but there's more to why he made his decision.

The quote mentioned in the question

To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret.

is from after Anakin helped Palpatine and Windu was killed as a result. First, at that point, it was a bit late for Anakin to backtrack and be like "Oh so you don't even know for sure how to do it? Then I'm out." and go back to the Jedi. Second, keep in mind that this was the first time Palpatine mentioned the ability to cheat death with much less degree of confidence than before.

At first, he only hinted that it's possible

— Darth Plagueis was a dark lord of the Sith so powerful and so wise, he could use the Force to influence the midi-chlorians to create life. He had such a knowledge of the dark side, he could even keep the ones he cared about from dying.

...

— Is it possible to learn this power?

— Not from a Jedi.

The next time they met, his statement was very confident.

Learn to know the dark side of the Force, and you will be able to save your wife from certain death.

As it was when Windu was about to kill Palpatine.

I have the power to save the one you love.

It means that Anakin thought that Palpatine already knew how to cheat death before making his choice. But saving Padme alone wasn't enough for him to betray the Jedi, even though he already had some doubts about them when the council asked him to spy on Palpatine.

You're asking me to do something against the Jedi code, against the Republic, against a mentor and a friend.

Even after that he still told Windu that Palpatine is a Sith as soon as he found it out, which almost certainly meant that Palpatine won't be around to help him save Padme. That was also the reason why he simply didn't have a chance to question Palpatine about his ability to cheat death. He couldn't accept him being a Sith back then. Next time they met, he obviously didn't have the time to ask questions.

Still Anakin wasn't sure if he's making the right choice, as Windu noted

For your own good, stay out of this affair. I sense a great deal of confusion in you, young Skywalker. There is much fear that clouds your judgement.

Only when Windu was going to kill an "unarmed" and "weak" old man, which was definitely against the Jedi teachings, he made the final choice. At that point, his disappointment in the Jedi was almost as good of a reason for him to support Palpatine as saving Padme.