Use the definition of the derivative to find f'(x) if f(x) = (1)/(x - 1) ​

Question

Use the definition of the derivative to find f'(x) if


`f(x) = (1)/(x - 1)`
​

Answer 1

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

Explanation:

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

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

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

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

=
`lim_(h>0)`
`(-h)/(h(x+h-1)(x-1))` ← cancel h on numerator denominator

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