By practicing with reviving for a little while, I've determined that the formula grows exponentially - every time you die, the base score for you to revive doubles. Now, that's affected by the level that your "Save Me" powerup is - that reduces the cost to revive. So, while it doesn't increase exactly exponentially, it is stunted somewhat. That's because once you've revived four or five times, it becomes significantly more expensive to revive, and increasingly unlikely to do so without the purchase of gems.
I've written up a short comparison table that compares the cost to revive, based on what revive attempt you're on in a single run.
The table above is based on if you have not levelled up the "Save Me" powerup. As you can see, the cost for a single revive (in column 2) is quite small. As you progress, the cost increases quite rapidly, until you're spending a large amount of gems in a single go.
I'll back be soon, with a fresh new post!
All the best, Josh