NCERT Class 12 · Ch 5 Exercise 5.3 · Q9
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.3 Question 9
Differentiate y=sin⁻¹(2x/(1+x²)).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: y′=2/(1+x²) for |x|<1; −2/(1+x²) for |x|>1; undefined at x=±1.
- Put u=2x/(1+x²). Then u′=2(1−x²)/(1+x²)² and √(1−u²)=|1−x²|/(1+x²).
- For x≠±1, y′=2(1−x²)/[(1+x²)|1−x²|]. This is positive inside (−1,1) and negative outside.
- At each of ±1 the one-sided slopes have opposite signs, so no derivative exists there.
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