Show that
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \cosh ^ { - 1 } \frac { 1 } { x } \right) = \frac { - 1 } { x \sqrt { 1 - x ^ { 2 } } }$$
(3 marks)
A curve has equation
$$y = \sqrt { 1 - x ^ { 2 } } - \cosh ^ { - 1 } \frac { 1 } { x } \quad ( 0 < x < 1 )$$
Show that:
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - x ^ { 2 } } } { x }\);
(4 marks)
the length of the arc of the curve from the point where \(x = \frac { 1 } { 4 }\) to the point where
$$x = \frac { 3 } { 4 } \text { is } \ln 3 .$$
(5 marks)