(i) Show that \(\frac{\mathrm{d}^{n+1}}{\mathrm{~d} x^{n+1}}\left(x^{n+1} \ln x\right)=\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x^{n}+(n+1) x^{n} \ln x\right)\).

(ii) Prove by mathematical induction that, for all positive integers \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x^{n} \ln x\right)=n!\left(\ln x+1+\frac{1}{2}+\ldots+\frac{1}{n}\right)\)