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Question
Differentiate the following w.r.t. x : `sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
Solution
Let y = `sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
= `sin^-1[(1 - (5x)^2)/(1 + (5x)^2)]`
Put 5x = tanθ.
Then θ = tan–1(5x)
∴ y = `sin^-1((1 - tan^2θ)/(1 + tan^2θ))`
= sin–1(cos2θ)
= `sin^-1[sin(pi/2 - 2θ)]`
= `pi/(2) - 2θ`
= `pi/(2) - 2tan^-1(5x)`
Differentiating w.r.t. x, we get
∴ `"dy"/"dx" = "d"/"dx"[pi/2 - 2tan^-1 (5x)]`
= `"d"/"dx"(pi/2) - 2"d"/"dx"[tan^-1(5x)]`
= `0 - 2 xx (1)/(1 + (5)^2)."d"/"dx"(5x)`
= `(-2)/(1 + 25x^2) xx 5`
= `(-10)/(1 + 25x^2)`.
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