- (i) Given that
$$x = \sec ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }$$
show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 x \sqrt { ( x - 1 ) } }$$
(ii) Given that
$$y = \left( x ^ { 2 } + x ^ { 3 } \right) \ln 2 x$$
find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \mathrm { e } } { 2 }\), giving your answer in its simplest form.
(iii) Given that
$$f ( x ) = \frac { 3 \cos x } { ( x + 1 ) ^ { \frac { 1 } { 3 } } } , \quad x \neq - 1$$
show that
$$\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { g } ( x ) } { ( x + 1 ) ^ { \frac { 4 } { 3 } } } , \quad x \neq - 1$$
where \(\mathrm { g } ( x )\) is an expression to be found.