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Question
Find the derivative of the following function from first principle.
`(x+1)/(x -1)`
Solution
`f(x) = (x + 1)/(x - 1)` and `f(x + h) = (x + h + 1)/(x + h - 1)`
`f(x + h) - f(x) = (x + 1 + h)/(x - 1 + h) - (x + 1)/(x - 1)`
= `((x + 1)(x - 1) + h(x - 1) - (x + 1)(x - 1) - h(x + 1))/((x - 1)(x - 1 + h))`
= `(h(x - 1 - x - 1))/((x - 1)(x - 1 + h))`
= `(-2h)/((x - 1)(x - 1 + h))`
`f'(x) = lim_(h → 0) (f(x + h) - f(x))/h`
= `lim_(h → 0) (-2h)/(h(x - 1)(x - 1 + h))`
= `(-2)/(x - 1)^2`
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