Question

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

• A. $2$
• B. $8$
• C. $16$
• D. $4$

Answer 1

The correct option is C $16$

$y=f\left(f\right(f\left(f\right(x\left)\right)\left)\right)$
Thus using chain rule
$y^{\prime }\left(x\right)=f^{\prime }\left(f\left(f\left(f\left(x\right)\right)\right)\right).f^{\prime }\left(f\left(f\left(x\right)\right)\right).f^{\prime }\left(f\left(x\right)\right).f^{\prime }\left(x\right)$
$\therefore y^{\prime }\left(0\right)=f^{\prime }\left(f\left(f\left(f\left(0\right)\right)\right)\right)f^{\prime }\left(f\left(f\left(0\right)\right)\right)f^{\prime }\left(f\left(0\right)\right)f^{\prime }\left(0\right)$
$=f^{\prime }\left(f\left(f\left(0\right)\right)\right)f^{\prime }\left(f\left(0\right)\right)f^{\prime }\left(0\right)f^{\prime }\left(0\right)$
$={\left(f^{\prime }\left(0\right)\right)}^4=2^4=16$