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

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