It is given that \(y=(1+x)^{2} \ln (1+x)\). Find \(\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}\).

Prove by mathematical induction that, for every integer \(n \geqslant 3\),
\(\frac{\mathrm{d}^{n} y}{\mathrm{~d} x^{n}}=(-1)^{n-1} \frac{2(n-3)!}{(1+x)^{n-2}}\)