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Question
Differentiate the following w.r.t. x : cosec–1 (e–x)
Solution
Let y = cosec–1 (e–x)
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"["cosec"^-1 (e^-x)]`
= `(-1)/(e^-x sqrt((e^-x)^2 - 1))."d"/"dx"(e^-x)`
= `(-1)/(e^-x sqrt(e^(-2x) - 1)) xx e^-x."d"/"dx"(– x)`
= `(-1)/sqrt(e^(-2x) - 1) xx -1`
= `(1)/(sqrt(1/e^(2x) - 1)`
= `e^x/(sqrt(1 - e^(2x))`
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