Question

If $f(1) = 1, f'(1) = 3$, then the derivative of $f(f(f(x))) + (f(x))^2$ at $x = 1$ is

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

Hint:

If $f(1) = 1$, then any nested layer $f(f(\dots f(1)\dots))$ also equals 1, simplifying the calculation significantly.

Answer 1

The Correct Option is D

Step 1: Concept
Use the Chain Rule for differentiation: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

Step 2: Meaning
Let $y = f(f(f(x))) + [f(x)]^2$. Then $\frac{dy}{dx} = f'(f(f(x))) \cdot f'(f(x)) \cdot f'(x) + 2f(x)f'(x)$.

Step 3: Analysis
At $x = 1$: $\frac{dy}{dx} = f'(f(f(1))) \cdot f'(f(1)) \cdot f'(1) + 2f(1)f'(1)$. Since $f(1) = 1$: $\frac{dy}{dx} = f'(f(1)) \cdot f'(1) \cdot f'(1) + 2(1)(3)$. $= f'(1) \cdot f'(1) \cdot f'(1) + 6 = (3 \cdot 3 \cdot 3) + 6$. $= 27 + 6 = 33$.

Step 4: Conclusion
The derivative at $x = 1$ is 33. Final Answer: (D)