The problem is:
Suppose that you have $f(z)=z^2-10 \in \mathbb{Q}(z)$ and denote by $f^3=f\circ f\circ f$. Define $$\phi(z)=\frac{f^3(z)-z}{f(z)-z}.$$ If $z$ and $w$ are roots of $\phi$, then $$[(f^3)'(z) - (f^3)'(w)]^2,$$ is rational.
I tried to resolve this but I couldn't. You can suppose that $f(z) - z$ divides $f^3(z) - z$ and $f(z)-z$ does not have shared root with $\phi$.