3. The curve \(C\) has equation
$$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
- Show that the point \(P ( 4 , - 8 )\) lies on \(C\).
- Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
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