The results of the Mathematica's code
f := Function[{n, x}, n^2*x^(n - 1) - D[(x^n - 1)/(x - 1), x]]
g := Function[{n, x},Sqrt[f[n, x]]/n + Sum[Sqrt[f[j, x]]/(j*(j + 1)), {j, 1, n - 1}]]
n = 60; FullSimplify[ g[n, x]^2 + Sum[(g[n, x] - g[k, x])^2, {k, 1, n}]]
60 x^59
n = 34; FullSimplify[g[n, x]^2 + Sum[(g[n, x] - g[k, x])^2, {k, 1, n}]]
34 x^33
suggest the answer $nx^{n-1}$. The same result is obtained with Maple 2016 through
restart; f := (n, x)-> n^2*x^(n-1)-(diff((x^n-1)/(x-1), x)):
g := (n, x) -> sqrt(f(n, x))/n+sum(sqrt(f(j, x))/(j*(j+1)), j = 1 .. n-1) :
simplify(expand([seq(sum((g(N, x)-g(k, x))^2, k = 1 .. N)+g(N, x)^2, N = 1 .. 5)]));
[1, 2x, 3x^2, 4x^3, 5x^4]