If y = tan⁻¹((√(1 + x²) − 1)/x), then y′(0) is:
If y = tan⁻¹((√(1 + x²) − 1)/x), then y′(0) is:
- A. 1/2
- B. −1/2
- C. 0
- D. 1
Answer: A) 1/2
Explanation: Put x = tan θ, then √(1+tan²θ) = sec θ. Expression = tan⁻¹((sec θ − 1)/tan θ) = tan⁻¹((1 − cos θ)/sin θ) = tan⁻¹(tan(θ/2)) = θ/2 = (tan⁻¹ x)/2. So y′ = 1/(2(1+x²)). At x=0, y′ = 1/2.
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