If `y=(f_(0)f_(0)f)(x)andf(0)=0,f'(0)=2` then `y'(0)` is equal to

Options

A. 6
B. 7
C. 8
D. 9

Answer

To solve the problem, we need to find the derivative \( y' \) at \( x = 0 \) for the function \( y = f(f(f(x))) \) given that \( f(0) = 0 \) and \( f'(0) = 2 \).

### Step-by-Step Solution:

1. **Identify the function**: We have \( y = f(f(f(x))) \).

2. **Differentiate using the chain rule**:
To differentiate \( y \) with respect to \( x \), we apply the chain rule multiple times:
\[ y' = f'(f(f(x))) \cdot f'(f(x)) \cdot f'(x) \]

3. **Evaluate at \( x = 0 \)**:
We need to find \( y'(0) \):
\[ y'(0) = f'(f(f(0))) \cdot f'(f(0)) \cdot f'(0) \]

4. **Substitute known values**:
We know \( f(0) = 0 \):
- Thus, \( f(f(0)) = f(0) = 0 \)
- And \( f(f(f(0))) = f(f(0)) = f(0) = 0 \)

Therefore, we can substitute these values into our derivative:
\[ y'(0) = f'(f(f(0))) \cdot f'(f(0)) \cdot f'(0) = f'(0) \cdot f'(0) \cdot f'(0) \]

5. **Substituting \( f'(0) \)**:
We know \( f'(0) = 2 \):
\[ y'(0) = 2 \cdot 2 \cdot 2 = 8 \]

6. **Final result**:
Therefore, the value of \( y'(0) \) is \( 8 \).

### Final Answer:
\[ y'(0) = 8 \]