Math problem thread

Find all differentiable functions f: R -> R+ for which the following is true: f'(x) = f(f(x))

Here's a nice one:


Before his death, an old pirate hid his treasure on an island with the following instructions for finding it later:
"On the island there is only one coconut palm tree, one banana tree and one big boulder. Start near the boulder, walk to the coconut tree, pick a coconut, turn 90 degrees counter-clockwise and walk then as many steps as you did from the boulder to the palm tree. Leave the coconut there. Return to the boulder, and walk this time to the banana tree, turning 90 degrees clockwise, and again walk as many steps as you did from the boulder to the banana tree. The treasure is now buried at the midpoint between you and the coconut."

You get your hands on these instructions and decide to get the treasure, so you set off to the island. You search around, but you can only find the coconut palm and the banana tree -- the boulder seems to be gone! The mission now seems impossible, but there is still a way to find the treasure. How?