5 The diagram shows part of a curve with a maximum point \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-06_472_791_358_630} The equation of the curve is $$y = 15 x ^ { \frac { 3 } { 2 } } - x ^ { \frac { 5 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the maximum point \(M\).
3. The point \(P ( 1,14 )\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20 x - 6\).
4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(R M\).