I would like to know how the big jump is possible (2:22 in the video) ? I play with a playstation controller.

You need two jump buttons. Either both on the controller, or just the jump button on the controller and spacebar on the keyboard. Press one of the buttons first and then press the other button on the very next frame. It's really hard and will take a while. I have not found a consistent setup for it yet. At first you can try just spamming both buttons for a while and you will probably make it after some time.

EDIT I was wondering why I kept making these "super jumps" (let's call them that :)) during the second half of the run, and it seems that the cutscene right after leaving the cave (with a pearl inside) triggers it. After watching that cutscene you can make these "super jumps" by just jumping once and then jumping again right as you land on the ground. It seems like it is a 2 or 3 frame window so not too hard. Definitely easier than doing what I described above. EDIT 2 It requires the feather (double jump) but no other item!

EDIT 3 I changed the route and now just skip the running shoes and no longer do the super jumps the way mentioned earlier (with two jump buttons). So you can skip having two jump buttons now.