Question

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

• A. 2
• B. 8
• C. 16
• D. 4

Answer 1

The correct option is C 16

$y=f\left(f\right(f\left(f\right(x\left)\right))$

$y^{\prime }\left(x\right)=f^{\prime }\left(f\right(f\left(f\right(x\left)\right)\left)\right)f^{\prime }\left(f\right(f\left(x\right)\left)\right)f^{\prime }\left(f\right(x\left)\right)f^{\prime }\left(x\right)$

$\Rightarrow y^{\prime }\left(0\right)=f^{\prime }\left(f\right(f\left(f\right(0\left)\right)\left)\right)f^{\prime }\left(f\right(f\left(0\right)\left)\right)f^{\prime }\left(f\right(0\left)\right)f^{\prime }\left(0\right)$

$\Rightarrow f^{\prime }\left(f\right(f\left(0\right)\left)\right)f^{\prime }\left(f\right(0\left)\right)f^{\prime }\left(0\right)f^{\prime }\left(0\right)$

$=f^{\prime }\left(f\right(0\left)\right)f^{\prime }\left(0\right)f^{\prime }\left(0\right)f^{\prime }\left(0\right)$

$=f^{\prime }\left(0\right)f^{\prime }\left(0\right)f^{\prime }\left(0\right)f^{\prime }\left(0\right)$

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