Class 12 Continuity And Differentiability · Long Answer3 Marksbranch_sensitive_differentiation
Class 12 Maths · Continuity And Differentiability · Long Answer #141
Differentiate u=arctan[(√(1+x²)−1)/x] for x≠0.
✅ Step-by-Step Solution
Final Answer: 1/[2(1+x²)]
- Rationalize the inner quotient: (√(1+x²)−1)/x=x/(√(1+x²)+1).
- Put θ=arctan x∈(−π/2,π/2). The quotient is tan(θ/2), using sinθ=x/√(1+x²) and cosθ=1/√(1+x²).
- Since θ/2 lies in the principal arctangent interval, u=½arctan x. Differentiate using (arctan x)′=1/(1+x²).
- Therefore u′=1/[2(1+x²)] for x≠0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Rationalize the inner quotient: (√(1+x²)−1)/x=x/(√(1+x²)+1).
- [1 mark] Put θ=arctan x∈(−π/2,π/2). The quotient is tan(θ/2), using sinθ=x/√(1+x²) and cosθ=1/√(1+x²).
- [1 mark] Since θ/2 lies in the principal arctangent interval, u=½arctan x. Differentiate using (arctan x)′=1/(1+x²).
- [0.5 marks] Therefore u′=1/[2(1+x²)] for x≠0.
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