Question

Question 15: Find dy/dx : y= sec¯¹(1/2x²-1), 0 < x < 1/√2

Class 12 - Math - Continuity and Differentiability

Answer 1

Refer to the attachment.
-----------------
Formulas used
•d/dx(cos-¹x) = -1/√1-x²
•cos2∅ = 2cos²∅-1

Answer 2

Given     y = Sec⁻¹ [ 1/(2x² - 1) ] ,       0 < x < 1/√2.         To find    dy/dx.

      Sec y = 1/(2x² - 1)
      Cos y  = 2x² - 1     --- (1)

DIfferentiate on both sides of (1).

      - Sin y * dy/dx  =  4 x 
                   dy/dx  =  - 4x / sin y
                             = - 4x / [ 2x √[1 - x²]
                             = - 2 / √(1-x²)         Answer


 As  sin y = √(1 - Cos² y) 
                = √ [ 1 - (2x² - 1)² ]   = √ [ (1 - 2x² + 1) (1 + 2x² - 1) ]    
               = √ [ 2 x² 2 (1 - x²) ]  = 2 x √(1-x²)