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Question
Differentiate the following w.r.t.x: `log(sqrt((1 - sinx)/(1 + sinx)))`
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Solution
Let y = `log(sqrt((1 - sinx)/(1 + sinx)))`
= `log(sqrt((1 - sinx)/(1 + sinx) xx (1 - sinx)/(1 - sinx)))`
= `log(sqrt((1 - sinx)^2/(1 - sin^2x)))`
= `log(sqrt((1 - sinx)^2/(cos^2x)))`
= `log((1 - sinx)/(cosx))`
= `log(1/cosx - sinx/cosx)`
= log(sec x – tan x)
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[log(secx - tanx)]`
= `(1)/(secx - tanx)."d"/"dx"(secx - tanx)`
= `(1)/(secx - tanx) xx (secx tanx - sec^2x)`
= `(-secx(secx - tanx))/(secx - tanx)`
= –sec x.
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