Sum

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.

Concept: Differentiation

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