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Question
Differentiate the following w.r.t. x : `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`
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Solution
Let y = `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`
= `tan^-1((1 + sqrt(x))/(1 - sqrt(x))) ...[∵ cot^-1 x = tan^-1(1/x)]`
= `tan^-1((1 + sqrt(x))/(1 - 1 xx sqrt(x)))`
= `tan^-1(1) + tan^-1(sqrt(x)) ...[∵ tan^-1((x + y)/(1 - xy)) = tan^-1x + tan^-1y]`
= `pi/(4) + tan^-1(sqrt(x))`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[pi/4 + tan^-1(sqrt(x))]`
= `"d"/"dx"(pi/4) + "d"/"dx"[tan^-1(sqrt(x))]`
= `0 + (1)/(1 + (sqrt(x))^2)."d"/"dx"(sqrt(x))`
= `(1)/(1 + x) xx (1)/(2sqrt(x)`
= `(1)/(2sqrt(x)(1 + x)`.
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