imo class 12 continuity and differentiability

If y = logₑ((x² + 1)/√(x² − 1)), then dy/dx is:

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If y = logₑ((x² + 1)/√(x² − 1)), then dy/dx is:

  • A. (x³ − 3x)/(x⁴ − 1)
  • B. (x³ + 3x)/(x⁴ − 1)
  • C. (3x − x³)/(x⁴ − 1)
  • D. (x³ − x)/(x⁴ − 1)

Answer: A) (x³ − 3x)/(x⁴ − 1)

Explanation: y = log(x²+1) − ½ log(x²−1). y′ = 2x/(x²+1) − (1/2)·2x/(x²−1) = 2x/(x²+1) − x/(x²−1) = (2x(x²−1) − x(x²+1))/((x²+1)(x²−1)) = (x³ − 3x)/(x⁴−1).

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