This game contains 6 life/health upgrades for the prince. This guide will show you where to get them all.

After you have finished playing as the dark prince for the first time and are the normal prince once again, climb until you are near two doorways. It's right after you fight enemies scared of light in dark rooms with light coming from the top. There is a dagger switch that opens a door there. After you go through this door you will have two paths. One on your left and one on your right. Left path leads to the life upgrade and the right path with blades leads to a save fountain and the next area.
The save fountain is called 'The Tunnels'. Since the fountain is so close, it might be worth saving first. It's right after the blades.

There is a large room right before the 1st boss - The giant guy with a sword. Once you activate the dagger switch, two doors will begin to open. The upgrade is in the door that is the farthest away from the lever. Go down that ramp and roll under the door before it closes. Continue down until you reach the fountain. The left, closer door leads to a save fountain called 'The Arena', but you can't go back so get the life upgrade first.

This is right after the save fountain called 'The Temple' which is right in front of a door with a dagger switch. When you go through and get to a balcony, you see the Vizier cut down one of Babylon's guards and you will become the dark prince. Then you fight and jump your way to a room, where you see the vizier fly up and leave. You will once again become the normal prince since the room is filled with water. This is the room where the upgrade is. Turn the rotating lever to drain the water and climb up the pole. Continue up, until you are on the swinging poles.
You should see a hole in the wall covered by a curtain. Make your way into that and continue to the fountain. If you go up instead, you will continue the story.

One of the bosses is a stone guardian that you fight in an arena. After that you ride him through a couple of doors. At the end, you break a huge door and he dies. You should see a ladder on a wall. Go up the ladder and there is a save fountain called 'The Canal'. Save and on the right you will see a hole. Roll through the hole and continue to the fountain.

I would say this one is the most annoying to get into. It's after the save fountain called 'The Palace Entrance'. You have to fix the elevator to get to the throne room. Each rotating lever moves the elevator up a bit. After the first one is done, you will get to a balcony with dancers and another rotating lever.
Dont use the rotating lever yet. From the balcony you should be able to see dagger plants and a button.

Jump up the dagger plants, wall jump to press the button and slow down the time. While falling, put your dagger into the middle dagger plant, so you can jump back onto the balcony and run into the door.

This one is right after the save fountain called 'The Middle Tower'. After the fountain, there is a dagger switch, that opens a door. There should be enemies outside of it. After climbing a little, you are at the second lift with a box, that you have to move onto elevators during the climb of the tower. The secret is up top, but you can't get there yet. You need to move the box and do a little bit of platforming.
Right before you go back into the tower, there is a wooden platform, where you will be prompted to use the landscape view. This will show you a way to get down to the area below. Following this, you will make your way to the fountain.

Hope this guide helped you.
Your health bar should now be full like this
Thank you for your attention and good luck.
If you have managed to beat the game without a single life upgrade then kudos to you!