Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f$ and $g$ be continuously differentiable functions such that $f(0) = 0, f'(0) = 2$ and $g(x) = f(-x + f(x))). The value of $g'(0)$ equals.
Step-by-Step Solution
Key Concept: Use the chain rule carefully when composing derivatives through nested functions.
Given $g(x) = f(-x + f(f(x)))$, differentiate using the chain rule: $g'(x) = f'(-x + f(f(x))) \cdot (-1 + f'(f(x)) \cdot f'(x))$. At $x = 0$, we have $g'(0) = f'(f(f(0))) \cdot (-1 + f'(f(0)) \cdot f'(0))$. Given $g'(0) = 6$, this determines the relationship between $f'$ values at specific points.
Correct Answer: 1