9 A function is defined by \(y = f ( t )\) where \(f ( t ) = \ln ( 1 + a t )\) and \(a\) is a constant.
- By considering \(\frac { d y } { d t } , \frac { d ^ { 2 } y } { d t ^ { 2 } } , \frac { d ^ { 3 } y } { d t ^ { 3 } }\) and \(\frac { d ^ { 4 } y } { d t ^ { 4 } }\), make a conjecture for a general formula for \(\frac { d ^ { n } y } { d t ^ { n } }\) in terms of \(n\) and \(a\) for any integer \(n \geqslant 1\).
- Use induction to prove the formula conjectured in part (a).
- In the case where \(\mathrm { f } ( t ) = \ln ( 1 + 2 t )\), find the rate at which the \(6 ^ { \text {th } }\) derivative of \(\mathrm { f } ( t )\) is varying when \(t = \frac { 3 } { 2 }\).