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Question
Differentiate the following w.r.t. x : `sin^-1(2xsqrt(1 - x^2))`
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Solution
Let y = `sin^-1(2xsqrt(1 - x^2))`
Put x = sinθ.
Then θ = sin–1x
∴ y = `sin^-1(2sinθsqrt(1 - sin^2θ))`
= sin–1(2sinθ cosθ)
= sin–1(sin2θ)
= 2θ
= 2sin–1x
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(2sin^-1x)`
= `2"d"/"dx"(sin^-1x)`
= `2 xx 1/sqrt(1 - x^2)`
= `2/sqrt(1 - x^2)`
We can also put x = cosθ.
Then θ = cos–1x
∴ y = `sin^-1(2cosθsqrt(1 - cos^2θ))`
= sin–1(2cosθ sinθ)
= sin–1(sin2θ)
= 2θ
= 2cos–1x
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(2cos^-1 x)`
= `2"d"/"dx"(cos^-1 x)`
= `2 xx (-1)/sqrt(1 - x^2)`
= `(-2)/sqrt(1 - x^2)`
Hence, `"dy"/"dx" = ± (2)/sqrt(1 - x^2)`.
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