Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f: [-1,1] \to \left[-\frac{\pi}{4}, \tan 1 + \tan^{-1}\right]$ defined by $f(x) = \tan x + \tan^{-1} x$ and the derivative of $f^{-1}(x)$ at $x = 0$ is $'k'$ then the value of $\frac{4}{k}$ is_____.
Step-by-Step Solution
Key Concept: The derivative of an inverse function equals the reciprocal of the derivative of the original function at the corresponding point.
The derivative of the inverse function at a point is $\left(f^{-1}(x)\right)' = \frac{1}{f'(f^{-1}(x))}$. At $x = 0$, this becomes $\frac{1}{f'(f^{-1}(0))} = \frac{1}{f'(0)}$.
Correct Answer: 0