If $f\left(0\right)=0,f^1\left(0\right)=2$ then the derivative of $y=f\left(f\left(f\left(x\right)\right)\right)$ at $x=0$

$\begin{array}{l}f\left(0\right)=0,f^1\left(0\right)=2\\ y=f\left(f\left(f\left(x\right)\right)\right)\\ \frac{d\left(y\right)}{dx}=\frac{d}{dx}\left[f\left(f\left(f\left(x\right)\right)\right)\right]\\ =f^1\left(f\left(f\left(x\right)\right)\right).f^1\left(f\left(x\right)\right).f^1\left(x\right)\\ =f^1\left(f\left(f\left(0\right)\right)\right).f^1\left(f\left(0\right)\right).f^1\left(0\right)\\ =f^1\left(0\right).f^1\left(0\right).2\\ =\mathrm{2.2.2}=8\end{array}$