Answer
$$F'(0)=96$$
Work Step by Step
$$F(x)=f(3f(4f(x))))$$
$$F'(x)=\frac{df(3f(4f(x))))}{dx}$$
According to Chain Rule: $$F'(x)=\frac{df(3f(4f(x))))}{d(3f(4f(x)))}\frac{3d(f(4f(x)))}{d(4f(x))}\frac{4df(x)}{dx}$$ $$F'(x)=f'(3f(4f(x)))\times3f'(4f(x))\times4f'(x)$$ $$F'(x)=12f'(3f(4f(x)))f'(4f(x))f'(x)$$
Therefore, $$F'(0)=12f'(3f(4f(0)))f'(4f(0))f'(0)$$
We know $f(0)=0$ and $f'(0)=2$ $$F'(0)=12f'(3f(4\times0))f'(4\times0)\times2$$ $$F'(0)=24f'(3f(0))f'(0)$$
Again, apply $f(0)=0$ and $f'(0)=2$ $$F'(0)=24f'(3\times0)\times2$$ $$F'(0)=48f'(0)$$
Finally, apply $f'(0)=2$ $$F'(0)=48\times2=96$$
