This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems.

I would like to know whether there exists a differentiable function from the (open or closed) unit interval to itself satisfying $$1-x-f(f(x))-f(x)f'(f(x))=0$$ for all $x$.

Ideally, I'd also like a list of all such functions.

This arose in the course of an economics problem whose description would be off-topic here, so lest this question seem too localized, let me pose a general question:

What techniques are available for solving functional equations of the above type?