Question

If $y=f\left(f\right(f\left(x\right)\left)\right),f\left(0\right)=0$ and $f^{\prime }\left(0\right)=2$ then find the value of $y^{\prime }\left(0\right)$

Answer 1

$y=f\left(f\right(f\left(x\right)\left)\right)\Rightarrow y^{\prime }=f^{\prime }\left(f\right(f\left(x\right)\left)\right).f^{\prime }\left(f\right(x\left)\right).f^{\prime }\left(x\right)$
$\therefore y^{\prime }\left(0\right)=f^{\prime }\left(f\right(f\left(0\right)\left)\right).f^{\prime }\left(f\right(0\left)\right).f^{\prime }\left(0\right)$
$=f^{\prime }\left(f\right(0\left)\right).f^{\prime }\left(0\right).2$
$=f^{\prime }\left(0\right).2\times 2=2\times 2\times 2=8$