Question

If \( f(0) = 0 \), \( f'(0) = 3 \), then the derivative of \( y = f(f(f(f(f(x))))) \) at \( x = 0 \) is:

• A. \( 16 \)
• B. \( 32 \)
• C. \( 81 \)
• D. \( 243 \)

Hint:

For nested functions like \( f(f(f(x))) \), apply the Chain Rule repeatedly, ensuring that each derivative is correctly multiplied at every level.

Answer 1

The Correct Option is D

Step 1: Understanding the Chain Rule Given the function: \[ y = f(f(f(f(f(x))))) \] To differentiate this function, we use the Chain Rule, which states: \[ \frac{dy}{dx} = f'(f(f(f(f(x))))) \cdot f'(f(f(f(x)))) \cdot f'(f(f(x))) \cdot f'(f(x)) \cdot f'(x) \] 
Step 2: Evaluating the Function at \( x = 0 \) Since \( f(0) = 0 \), we can substitute: \[ y = f(f(f(f(f(0))))) = f(f(f(f(0)))) = f(f(f(0))) = f(f(0)) = f(0) = 0. \] Therefore, all occurrences of \( f(x) \) simplify to \( f(0) \), so: \[ \frac{dy}{dx} = f'(0) \cdot f'(0) \cdot f'(0) \cdot f'(0) \cdot f'(0) \] 
Step 3: Computing the Final Derivative Value Given that \( f'(0) = 3 \), we substitute: \[ \frac{dy}{dx} = 3 \times 3 \times 3 \times 3 \times 3 \] \[ = 3^5 = 243 \] 
Final Answer: \[ \frac{dy}{dx} = 243 \] Thus, the correct option is:  Option (4): 243