Q104 / 160 · AP EAMCET 2024 (22 May Shift 1) · Mathematics
If $f(0)=0, f^{\prime}(0)=3$, then the derivative of $y=f(f(f(f(f(x)))))$ at $x=0$ is
- A16
- B32
- C81
- Correct answerD243
Solution
$\begin{aligned} & f(0)=0, f^{\prime}(0)=3 \\ & y=f(f(f(f(f(x))))) \\ & \frac{d y}{d x}=f^{\prime}(f(f(f(f(x))))) f^{\prime}(f(f(f(x))))\end{aligned}$
$f^{\prime}(f(f(x))) f^{\prime}(f(x)) f^{\prime}(x)$
$\begin{array}{r}\left.\frac{d y}{d x}\right|_{x=0}=f^{\prime}(f(f(f(f(0))))) f^{\prime}(f(f(f(0)))) \\ f^{\prime}(f(f(0))) f^{\prime}(f(0)) f^{\prime}(0)\end{array}$
$=\left[f^{\prime}(0)\right]^5=3^5=243$.
$\begin{array}{r}\left.\frac{d y}{d x}\right|_{x=0}=f^{\prime}(f(f(f(f(0))))) f^{\prime}(f(f(f(0)))) \\ f^{\prime}(f(f(0))) f^{\prime}(f(0)) f^{\prime}(0)\end{array}$
$=\left[f^{\prime}(0)\right]^5=3^5=243$.
Hence, the correct answer is option: 'D'.