1.

यदि ` y= sin (tan ^(-1) 2x )` तब सिद्ध कीजिए की `(dy)/(dx)= (2)/((1+4x^(2) )^(3//2) ) `

Answer» `y= sin (tan^(-1) 2x ) `
माना ` tan ^(-1) 2x=t`
` therefore " "y= sin t `
` rArr " "(dy)/(dt) =cos t ` व ` (dt)/(dx) ={ (1)/((1+ 4x^(2) ))*2} =(2)/((1+4x^(2) ) ) `
हम जानते है की
` " "(dy)/(dx) =((dy)/(dt)xx(dt)/(dx))`
` (dy)/(dt) ` व ` (dt)/(dx) ` के मान रखने पर
` " "(dy)/(dx) ={ cos txx (2)/((1+4x^(2) ))}" "...(1) `
` t= tan ^(-1) 2xrArr tan t =2x `
` rArr " "sec t = sqrt (1+tan ^(2) t )= sqrt ( 1+ 4x^(2) )`
` rArr " " cost = (1)/(sqrt (1+ 4x^(2) ) )`
समीकरण (1 ) में cos t का मान रखने पर
` (dy)/(dx) ={ (1)/(sqrt (1+4x ^(2)))xx(2)/((1+ 4x^(2))) } =(2)/((1+4x^(2))^(3//2) ) `


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