Prove that, if \(y = \sin ^ { - 1 } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
Find the Maclaurin series for \(\sin ^ { - 1 } x\), up to and including the term in \(x ^ { 3 }\).
Use the result of part (ii) and the Maclaurin series for \(\ln ( 1 + x )\) to find the Maclaurin series for \(\left( \sin ^ { - 1 } x \right) \ln ( 1 + x )\), up to and including the term in \(x ^ { 4 }\).