Method of Differentiation
Inverse Trigonometric Functions
JEE Main 2021
Grade 12
Question:
For $x \in (-1, 1)$, $\frac{d}{dx} \left[\sin^{-1}\left(\frac{2x}{1 + x^2}\right)\right]$ equals:
(1) $\frac{1}{1 + x^2}$
(2) $\frac{2}{1 + x^2}$
(3) $\frac{2x}{1 + x^2}$
(4) $\frac{1 - x^2}{1 + x^2}$
Step-by-Step Solution
Key Concept: Put $x = \tan \theta$ so that $\frac{2x}{1 + x^2} = \sin 2\theta$; the function then reduces to $2 \tan^{-1} x$ on the relevant interval.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)