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Question
Find the derivatives of the following:
`cos[2tan^-1 sqrt((1 - x)/(1 + x))]`
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Solution
Let y = `cos[2tan^-1 sqrt((1 - x)/(1 + x))]`
Put x = `cos 2theta`
`("d"x)/("d"theta) = - sin 2theta xx 2`
`("d"x)/("d"theta) = - 2 sin 2theta` .......(1)
y = `cos[2tan^- sqrt((1 - cos 2theta)/(1 + cos theta))]`
y = `cos[2tan^-1 sqrt((2sin^2theta)/(2cos^2theta))]`
y = `cos[2tan^-1 sqrt(tan^2theta)]`
y = `cos[2tan^-1 (tan theta)]`
y = `cos[2theta]`
`("d"y)/("d"theta) = - sin 2theta xx 2`
`("d"y)/("d"theta) = - 2 sin 2theta` .......(2)
From equation (1) and (2) we get
`(("d"y)/("d"theta))/(("d"x)/("d"theta)) = (- 2sin 2theta)/(- 2 sin 2theta)`
`("d"y)/("d"x)` = 1
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