Question

n 19. Iff(!) = 1, f'(1) = 5, then the derivative of f(f(f(x))) + (f(x))² at x = (1) is

• A. 125
• B. 1250
• C. 135

Answer 1

Answer: (C)

135 (Option c)

Answer 2

Explanation:
Given $f(1) = 1$ and $f'(1) = 5$.

1. Derivative of $f(f(f(x)))$ by chain rule at $x=1$: $$f'(f(f(1))) \cdot f'(f(1)) \cdot f'(1) = f'(1)^3 = 5^3 = 125.$$
2. Derivative of $(f(x))^2$ at $x=1$: $$2f(1)f'(1) = 2 \cdot 1 \cdot 5 = 10.$$
3. Total derivative: $$125 + 10 = 135.$$

Answer: 135 (Option c)