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 

Answer 2

To find the derivative of \( y = f(f(f(f(f(x))))) \) at \( x = 0 \), we'll apply the chain rule multiple times. Given that \( f(0) = 0 \) and \( f'(0) = 3 \), we will calculate \( y'(x) \) when \( x = 0 \).

The function is composed of 5 layers of \( f \), so each derivative step involves chaining these derivatives.

Let's denote:

• \( g(x) = f(f(f(f(f(x))))) \)
• \( h(x) = f(f(f(f(x)))) \)
• \( i(x) = f(f(f(x))) \)
• \( j(x) = f(f(x)) \)
• \( k(x) = f(x) \)

Each derivative follows from the chain rule:

1. \( g'(x) = f'(h(x)) \cdot h'(x) \)
2. \( h'(x) = f'(i(x)) \cdot i'(x) \)
3. \( i'(x) = f'(j(x)) \cdot j'(x) \)
4. \( j'(x) = f'(k(x)) \cdot k'(x) \)
5. \( k'(x) = f'(x) \)

At \( x=0 \):

• \( k(0) = f(0) = 0 \)
• \( j(0) = f(k(0)) = f(0) = 0 \)
• \( i(0) = f(j(0)) = f(0) = 0 \)
• \( h(0) = f(i(0)) = f(0) = 0 \)
• \( g(0) = f(h(0)) = f(0) = 0 \)

Using the chain rule and substituting known derivatives:

• \( g'(0) = f'(h(0)) \cdot h'(0) = f'(0) \cdot h'(0) \)
• \( h'(0) = f'(i(0)) \cdot i'(0) = f'(0) \cdot i'(0) \)
• \( i'(0) = f'(j(0)) \cdot j'(0) = f'(0) \cdot j'(0) \)
• \( j'(0) = f'(k(0)) \cdot k'(0) = f'(0) \cdot f'(0) = (f'(0))^2 \)

Thus, we compile all these:

\( g'(0) = (f'(0))^5 = 3^5 = 243 \)

So, the derivative of \( y = f(f(f(f(f(x))))) \) at \( x = 0 \) is \( \boxed{243} \).