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Question
Differentiate the following w.r.t. x :
`cos^-1((1 - x^2)/(1 + x^2))`
Solution
Let y = `cos^-1((1 - x^2)/(1 + x^2))`
Put x = tanθ.
Then θ = tan–1x
∴ y = `cos^-1((1 - tan^2θ)/(1 + tan^2θ))`
= cos–1(cos2θ)
= 2θ
= 2tan–1x
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(2tan^-1 x)`
= `2"d"/"dx"(tan^-1 x)`
= `2 xx (1)/(1 + x^2)`
= `(2)/(1 + x^2)`.
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