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Question
Find the derivatives of the following:
Find the derivative of `sin^-1 ((2x)/(1 + x^2))` with respect to `tan^-1 x`
Solution
Let u = `sin^-1 ((2x)/(1 + x^2))`
Put x = tan θ
u = `sin^-1 ((2tantheta)/(1 + tan^2theta))`
u = `sin^-1 (sin 2theta)`
u = 2θ
u = `2 tan^-1 (x)`
`("du")/("d"x) = 2/(1 + x^2)` ...(1)
Let v = `tan^-1 x`
`("dv")/("d"x) = 2/(1 + x^2)` ...(2)
From equations (1) and (2)
`(("du")/(dx))/(("dv")/(dx)) = (2/(1 + x^2))/(1/(1 + x^2))`
`("du")/("dv")` = 2
`("d"(sin^-1 ((2x)/(1 + x^2))))/("d"(tan^-1 x))` = 2
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