Given that \(x = \frac { \sin y } { \cos y }\), use the quotient rule to show that
$$\frac { \mathrm { d } x } { \mathrm {~d} y } = \sec ^ { 2 } y$$
(3 marks)
Given that \(\tan y = x - 1\), use a trigonometrical identity to show that
$$\sec ^ { 2 } y = x ^ { 2 } - 2 x + 2$$
Show that, if \(y = \tan ^ { - 1 } ( x - 1 )\), then
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } - 2 x + 2 }$$
(l mark)
A curve has equation \(y = \tan ^ { - 1 } ( x - 1 ) - \ln x\).
Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Hence show that the curve has a minimum point which lies on the \(x\)-axis.