Question

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

• A. $2$
• B. $8$
• C. $16$
• D. $4$

Answer 1

The correct option is C $16$

$y=f\left(f\right(f\left(f\right(x\left)\right))\Rightarrow y^{\prime }=f^{\prime }(f\left(f\right(f\left(x\right)\left)\right)\left)f^{\prime }\right(f\left(f\right(x\left)\right)\left)f^{\prime }\right(f\left(x\right)\left)f^{\prime }\right(x)$
Putting $x=0$
$\Rightarrow y^{\prime }=f^{\prime }\left(f\right(f\left(f\right(0\left)\right)\left)\right)f^{\prime }\left(f\right(f\left(0\right)\left)\right)f^{\prime }\left(f\right(0\left)\right)f^{\prime }\left(0\right)\Rightarrow y^{\prime }=f^{\prime }\left(0\right)f^{\prime }\left(0\right)f^{\prime }\left(0\right)f^{\prime }\left(0\right)\Rightarrow y^{\prime }=2^4=16$