Question

How do you find f'(x) using the definition of a derivative for `f(x)=(x+1)/(x-1)`?

Answer 1

We have:

`f'(x) = lim_(h->0) (f(x + h) - f(x))/h`

`f'(x) = lim_(h-> 0) ((x+ h + 1)/(x+ h - 1) - (x + 1)/(x -1))/h`

`f'(x) = lim_(h->0) ((x+ h + 1)(x -1) - ((x + 1)(x + h - 1)))/(h(x + h - 1)(x - 1))`

`f'(x) = lim_(h->0) ((x^2 + hx + x - x - h - 1 - (x^2 + xh - x + x + h - 1)))/(h(x + h - 1)(x- 1))`

`f'(x) = lim_(h->0) (-2h)/(h(x + h - 1)(x - 1))`

`f'(x) = -2/(x - 1)^2`

We woudl have obtained the same answer using the quotient rule.

Hopefully this helps!