Question

Find dy/dx if y = sin-1(xsqrt(1-x)-sqrt(x(1-x^2)))

Answer 1

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solution:--

∵ y = sinֿ¹ [ x.√(1-x) - √x. √(1-x²) ] 

∴ y = sinֿ¹ [ (x). √(1-(√x)²) - √x. √(1-(x)²) ] ....... (1) 
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Let : (x) = sin α and √x = sin ß. .................... (2) 

Then, from (1), 

y = sinֿ¹ [ sin α. √(1-sin² ß) - sin ß. √(1-sin² α) ] 

. = sinֿ¹ [ sin α. cos ß - sin ß. cos α ] 

. = sinֿ¹ [ sin ( α+ß ) ] 

. = α + ß 

. = sinֿ¹ x + sinֿ¹ √x ............. from (2) 
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∴ dy/dx = [ 1 / √(1-x²) ] + [ 1 / √(1-(√x)²) ]· d/dx ( √x ) 

. . . . . . = [ 1 / √(1-x²) ] + [ 1 / √(1-x) ]· [ 1 / (2√x) ] 

. . . . . . .= [ 1 / √(1-x²) ] + { 1 / [ 2√(x-x²) ] } 

Answer 2

Answer:

Refer to the attachment

Hope it helps you