Fractional part integral representation for derivatives of a function related to ln Gamma(x+1)

For $0\mathrm{\ne }x>-1$ let $$\mathrm{\Delta }(x)=\frac{\ln \mathrm{\Gamma }(x+1)}{x}\mathrm{.}$$ Recently Adell and Alzer proved the complete monotonicity of ${\mathrm{\Delta }}^{\mathrm{\prime }}$ on $(-1,\mathrm{\infty })$ by giving an integral representation of $(-1{)}^n{\mathrm{\Delta }}^{(n+1)}(x)$ in terms of the Hurwitz zeta function $\zeta (s,a)$. We reprove this integral representation in different ways, and then re-express it in terms of fractional part integrals. Special cases then have explicit evaluations. Other relations for ${\mathrm{\Delta }}^{(n+1)}(x)$ are presented, including its leading asymptotic form as $x \to \infty$.