no need to get lost, the important thing is to bring the expression to a convenient form:

f(x) = x(x^2-6x+16)/(x-2)^3 = (x^3 - 6x^2 + 16x) / (x-2)^3 =

= (x^3 -6x^2 + 16x)*(x-2)^-3

now as you know d/dx(a(x)*b(x)) = a '(x)*b(x) + a(x)*b '(x)

in the present case a(x) = (x^3 -6x^2 + 16x) , b(x) = (x-2)^-3

f '(x) = (3x^2-12x+16)/(x-2)^3 -3*(x^3 - 6x^2 + 16x)/(x-2)^4