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Question
Differentiate `sec^-1 (1/sqrt(1 - x^2))` w.r.t. `sin^-1 (2xsqrt(1 - x^2))`.
Solution
Let u = `sec^-1 1/sqrt(1 - x^2)`
x = sin θ
`\implies 1/sqrt(1 - x^2) = 1/sqrt(1 - sin^2θ)`
= `1/cosθ`
= sec θ
So u = sec–1 sec θ
= θ
= sin–1 x
`(du)/dx = 1/sqrt(1 - x^2)`
v = `sin^-1 (2xsqrt(1 - x^2))` ...(i)
Suppose that
x = sin α
v = `sin^-1 [2 sin αsqrt(1 - sin^2α)]`
= sin–1 [2 sin α . cos α]
= sin–1 [sin 2α]
= 2α
= 2 sin–1 x
`(dv)/dx = 2/sqrt(1 - x^2)` ...(ii)
From (i) and (ii)
∴ `(du)/(dv) = 1/2`
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