Is there a function satisfying the following properties $ f^{n}(x)=(f(x))^{n+1}$?? Is there a function with the following properties?
$$ f(x)=f(x) $$
$$ f'(x)=f(x)^2 $$
$$f^{(n)}(x)=\left(f(x)\right)^{n+1}$$
where $f^{(n)}$ denotes the $n$th derivative, and by convention $f^{(0)}(x) = f(x)$.

The function $x \mapsto 0: \mathbb{R} \to \mathbb{R}$ is one such.
