It is given that \(y=\ln (a x+1)\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\),
\(\frac{\mathrm{d}^{n} y}{\mathrm{~d} x^{n}}=(-1)^{n-1} \frac{(n-1)!a^{n}}{(a x+1)^{n}}\)