Write down the derivative of \(\sqrt { y ^ { 2 } + 1 }\) with respect to \(y\).
Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( x - 1 ) \sqrt { y ^ { 2 } + 1 } } { x y }\) and that \(y = \sqrt { \mathrm { e } ^ { 2 } - 2 \mathrm { e } }\) when \(x = \mathrm { e }\),
find a relationship between \(x\) and \(y\).