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प्रश्न
Differentiate the following w.r.t. x : `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`
उत्तर
Let y = `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`
Put x = cosθ. Thenθ = cos–1x and
`sqrt((1 - x)/(1 + x)) = sqrt((1 - cosθ)/(1 + cosθ)`
= `sqrt((2sin^2(θ/2))/(2cos^2(θ/2)`
= `sqrt(tan^2(θ/2)`
= `tan(θ/2)`
∴ `tan^-1(sqrt((1 - x)/(1 + x)))`
= `tan^-1[tan(θ/2)]`
= `θ/(2)`
= `(1)/(2)cos^-1 x`
∴ y = `sin[2 xx 1/2 cos^-1 x]`
= sin (cos–1x)
∴ `"dy"/"dx" = "d"/"dx"[sin(cos-1x)]`
= `cos(cos^-1x)."d"/"dx"(cos^-1x)`
= `x xx (-1)/sqrt(1 - x^2)`
= `(-x)/sqrt(1 - x^2)`.
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