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If y=sqrt((1-x)/(1+x)), find (dy)/(dx) a...

If `y=sqrt((1-x)/(1+x))`, find `(dy)/(dx)` and prove that `(1-x^2)(dy)/(dx)+y=0`

Text Solution

Verified by Experts

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

Multiplying with its conjugate to make denominator perfect square

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

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

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

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

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

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

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