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If y=sqrt((1+e^x)/(1-e^x)),s howt h a t...

If `y=sqrt((1+e^x)/(1-e^x)),s howt h a t(dy)/(dx)=(e^x)/((1-e^x)sqrt(1-e^(2x)))`

Text Solution

Verified by Experts

Given that,
`y=sqrt((1+e^x)/(1-e^x))`

Multiplying with its conjugate to make denominator perfect square

`y=sqrt((1+e^x)/(1-e^x)) times sqrt((1+e^x)/(1+e^x))`

`y=((1+e^x)/(sqrt(1-e^(2x))))`

`frac(dy}{dx}=(e^x sqrt(1-e^(2x))-(1+e^x)(-2e^(2x))/(2sqrt(1-e^(2x))))/(1-e^(2x))`

`frac(dy}{dx}=(e^x (1-e^(2x))+(1+e^x)(e^(2x)))/(1-e^(2x)) (1)/(sqrt(1-e^(2x))`

`frac(dy}{dx}=(e^x -e^(3x)+e^(2x)+e^(3x))/(1-e^(2x)) (1)/(sqrt(1-e^(2x))`

`frac(dy}{dx}=frac{e^x(1+e^(x))}{(1-e^x)(1+e^x)}(1)/(sqrt(1-e^(2x)) `

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