What is the derivative y = (sqrt x + 1)/(sqrt x -1)?

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The derivative of `y = (sqrt x + 1)/(sqrt x -1)` has to be determined.

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

Multiply the numerator and denominator by `sqrt x + 1`

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

=> `y = (x + 2*sqrt x + 1)/(x - 1)`

Use the quotient rule to find the derivative.

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

=> `y' = ((1 + 2*(1/2)*1/sqrt x)*(x - 1) - (x + 2*sqrt x + 1)*1)/(x - 1)^2`

=> `y' = (x - 1 + (x - 1)/sqrt x - x - 2*sqrt x - 1)/(x - 1)^2`

=> `y' = (-2 + (x - 1)/sqrt x - 2*sqrt x)/(x - 1)^2`

=> y' = `(-2*sqrt x + (x - 1) - 2x)/((x - 1)^2*sqrt x)`

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

The derivative of `y = (sqrt x + 1)/(sqrt x -1)` is `y' = (-2*sqrt x - x - 1)/((x - 1)^2*sqrt x)`

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