Question

If $f\left(0\right)=0,f^{\prime }\left(0\right)=3$ then the derivative of $y=f\left(f\right(f\left(x\right)\left)\right)$ at $x=0$ is

• A. $2$
• B. $81$
• C. $8$
• D. $4$

Answer 1

The correct option is B $81$

$y=f\left(f\left(f\left(x\right)\right)\right)y^{\prime }\left(x\right)=f^{\prime }\left(f\left(f\left(f\left(x\right)\right)\right)\right)f^{\prime }\left(f\left(f\left(x\right)\right)\right)f^{\prime }\left(f\left(x\right)\right)f^{\prime }\left(x\right)y^{\prime }\left(0\right)=f^{\prime }\left(f\left(f\left(f\left(0\right)\right)\right)\right)f^{\prime }\left(f\left(f\left(0\right)\right)\right)f^{\prime }\left(f\left(0\right)\right)f^{\prime }\left(0\right)=f^{\prime }\left(f\left(f\left(0\right)\right)\right)f^{\prime }\left(f\left(0\right)\right)f^{\prime }\left(0\right)f^{\prime }\left(f\right)={\left(f^{\prime }\left(0\right)\right)}^4=3^4=81$