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