6 It is given that \(y = \ln ( 1 + \sin x )\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
- Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \mathrm { e } ^ { - y }\).
- Express \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) in terms of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\mathrm { e } ^ { - y }\).
- Hence, by using Maclaurin's theorem, find the first four non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( 1 + \sin x )\).