1.

`(dy)/(dx)` ज्ञात कीजिए : `t=cos^(-1)((2x)/(1+x^(2))),-1 lt x lt 1`

Answer» `y=cos^(-1)((2x)/(1+x^(2)))" "{:("माना "x=tan theta),(rArr theta= tan^(-1)x):}`
`rArr" "y=cos^(-1)((2 tan theta)/(1+tan^(2)theta))`
`=cos^(-1)((2 tan theta)/(sec^(2) theta))=cos^(-1)(2 sin theta cos theta)`
`=cos^(-1)(sin 2 theta)=cos^(-1)cos((pi)/(2)-2 theta)`
`=(pi)/(2)-2 theta=(pi)/(2)-2 tan^(-1)x`
`rArr" "(dy)/(dx)=(d)/(dx)((pi)/(2)-2 tan^(-1)x)`
`=0-(2xx1)/(1+x^(2))=-(2)/(1+x^(2))`


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