It is given that
\(x=t+\sin t, \quad y=t^{2}+2 \cos t,\)
where \(-\pi\lt t\lt \pi\). Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) in terms of \(t\).

Show that
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{2 t \sin t}{(1+\cos t)^{3}} .\)

Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) increases with \(x\) over the given interval of \(t\).