(a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
\(1-\tanh ^{2} u=\operatorname{sech}^{2} u .\)
(b) Show that \(\frac{\mathrm{d}}{\mathrm{d} t}\left(\operatorname{sech}^{-1} t\right)=-\frac{1}{t \sqrt{1-t^{2}}}\).

It is given that
\(x=\tanh ^{-1} t \quad \text { and } \quad y=t \operatorname{sech}^{-1} t, \quad \text { for } 0<t<1 .\)
(c) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\sqrt{1-t^{2}}+\left(1-t^{2}\right) \operatorname{sech}^{-1} t\).
(d) Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in terms of \(t\).