For n=1, d/dx (x^-1)=-x^-2
n=2, d^2/dx^2 (x^-1)=2x^-3=2.1.x^-3
n=3, d^3/dx^3 (x^-1)=-6x^-4=-(3.2.1)x^-4 where d^3/dx^3 notates the 3rd derivative of (x^-1)
n=4, d^4/dx^4 (x^-1)=24x^-5=4.3.2.1x^-5
...
n=n, d^n/dx^n (x^-1)= [(-1)^n.n!]/(x^(n+1)]
Seems pretty obvious just via...