Question

प्रथम सिद्धांत से निम्नलिखित फलन का अवकलज ज्ञात कीजिए:

`(x + 1)/(x - 1)`

Answer 1

`f(x) = (x + 1)/(x - 1)` और `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`