Question

If $f\left(1\right)=1,f^{\prime }\left(1\right)=3$, then the derivative of $f\left(f\right(f\left(x\right)\left)\right)+\left(f\right(x){)}^2$ at $x=1$ is :

• A. $9$
• B. $12$
• C. $15$
• D. $33$

Answer 1

The correct option is D $33$

Let $y=f\left(f\right(f\left(x\right)\left)\right)+\left(f\right(x){)}^2$
$\Rightarrow \frac{dy}{dx}=f^{\prime }\left(f\right(f\left(x\right)\left)\right)\cdot f^{\prime }\left(f\right(x\left)\right)\cdot f^{\prime }\left(x\right)+2f\left(x\right)f^{\prime }\left(x\right)$
$\Rightarrow \frac{dy}{dx}{|}_{x=1}=f^{\prime }\left(f\right(f\left(1\right)\left)\right)\cdot f^{\prime }\left(f\right(1\left)\right)\cdot f^{\prime }\left(1\right)+2f\left(1\right)f^{\prime }\left(1\right)$
$=3\cdot 3\cdot 3+2\cdot 1\cdot 3=33$