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

1 Answer
May 22, 2017

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.