reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem Th29:
  x <> 0 implies ( #Z n)^ is_differentiable_in x & diff(( #Z n)^,x
  ) = -(n * x #Z (n-1))/(x #Z n)^2
proof
A1: ( #Z n).x = x #Z n & x #Z n = x|^n by PREPOWER:36,TAYLOR_1:def 1;
  assume x<>0;
  then
A2: ( #Z n).x <>0 by A1,PREPOWER:5;
A3: #Z n is_differentiable_in x by TAYLOR_1:2;
  then diff(( #Z n)^,x) = - diff( #Z n,x)/(( #Z n).x)^2 by A2,FDIFF_2:15
    .= -(n * x #Z (n-1))/( #Z n.x)^2 by TAYLOR_1:2
    .= -(n * x #Z (n-1))/(x #Z n)^2 by TAYLOR_1:def 1;
  hence thesis by A2,A3,FDIFF_2:15;
end;
