Question

Given f(x)=(2x+1)/(x-2) ,and x does not equal to 2.. find f^11(x)??

Answer 1

`f^((11))(x) = -5*(11!)*(x-2)^-12`

Explanation:

`f'(x) = -5/(x-2)^2` `" "` (by the quotient rule)

`= -5(x-2)^-2`

`f''(x) = 5(2)(x-2)^-3` `" "` (by power and chain rule)

`f^((3)) = -5(3*2)(x-2)^-4`

`f^((4)) = 5(4*3*2)(x-2)^-5`

In general

`f^((n)) = (-1)^n 5(n*(n-1)* * * 3*2)(x-2)^-(n+1)`

`= (-1)^n 5 (n!) (x-2)^-(n+1)`

So,

`f^((11))(x) = -5*(11!)*(x-2)^-12`

Note
`n! = n(n-1)(n-1) * * * 2*1` for positive integer `n`.