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Differentiate sqrt((1-tanx)/(1+tanx)) w....

Differentiate `sqrt((1-tanx)/(1+tanx))` w.r.t. x.

A

` (-sec^(2)x)/((1+tanx)^(3//2) (1-tanx))`

B

` (-sec^(2)x)/ (1-tanx)^(1//2)`

C

` (-secx)/((1+tanx)^(3//2) (1-tanx)^(1//2))`

D

` (-sec^(2)x)/((1+tanx)^(3//2) (1-tanx)^(1//2))`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `y = sqrt((1-tanx)/(1+tanx))`
Putting `(1-tanx)/(1+tanx) = t`, we get `y = sqrt(t)` and `t = (1-tanx)/(1+tanx)`
`:. (dy)/(dt) = (1)/(2) t^(-1//2) = 1/(2sqrt(t))`.
And, `(dt)/(dx) = ((1+tanx).(d)/(dx) (1-tanx) - (1-tanx).(d)/(dx) (1+tanx))/((1+tanx)^(2))`
`= ((1+tanx) (-sec^(2)x)-(1-tanx)(sec^(2)x))/((1+tanx)^(2)) = (-2sec^(2)x)/((1+tanx)^(2))`
`:. (dy)/(dx) = ((dy)/(dt) xx (dt)/(dx)) = (1)/(2sqrt(t)) xx (-2sec^(2)x)/((1+tanx)^(2))`
`= (-sec^(2)x)/((1+tanx)^(2)) xx (sqrt(1+tanx))/(sqrt(1-tanx))`
`= (-sec^(2)x)/((1+tanx)^(3//2) (1-tanx)^(1//2))`
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