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Differentiate

if y= xsin-1x /(1-x2 )1/2

y = xsin-1x1-x212⇒y = xsin-1x×1-x2-12⇒dydx = ddxxsin-1x×1-x2-12              = sin-1x×1-x2-12×ddxx+xddxsin-1x×1-x2-12    {using the product rule}              = sin-1x1-x2×1+xsin-1xddx1-x2-12+1-x2-12ddxsin-1x          = sin-1x1-x2+xsin-1x×d1-x2-12d1-x2×d1-x2dx+11-x2×11-x2   {Since; ddxsin-1x = 11-x2}         = sin-1x1-x2+xsin-1x×-121-x2-12-1×0-2x+11-x2            = sin-1x1-x2+xsin-1x×-121-x2-32×-2x+11-x2            = sin-1x1-x2+xxsin-1x1-x232+11-x2           = sin-1x1-x2+x2sin-1x1-x232+x1-x2             = 1-x2sin-1x+x2sin-1x+x1-x21-x232                = sin-1x-x2sin-1x+x2sin-1x+x1-x21-x232              = sin-1x+x1-x21-x21-x2Therefore; dydx = sin-1x+x1-x21-x21-x2

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cos-1/√(2x)

  • 29

cos-1/√(2x)

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x+(1x)2√sin1x(1x2(1x)2√)(1x)2

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frac{x + sqrt{(1 - x)^2}sin^{-1}x( 1 - x^2 sqrt{(1 - x)^2} ) }{(1 - x)^2}

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