91Ó°ÊÓ

From the definitions: \(\sinh \mathrm{x}=\left[\left(\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}\right) /(2)\right]\) \(\operatorname{Cos} x=\left[\left(e^{x}-e^{-x}\right) /(2)\right], \tanh x=\left[\left(e^{x}-e^{-x}\right) /\left[\left(e^{x}+e^{-x}\right)\right]\right.\) and Sech \(\mathrm{x}=\left[(2) /\left(\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{\mathrm{x}}\right)\right]\) (a) \(D_{x}(\sinh x)=\cosh x\) (b) \(D_{x}(\cosh x)=\sinh x\) (c) \(D_{x}(\tanh x)=\operatorname{sech}^{2} x\)

From the fact that \(\sinh ^{-1} \mathrm{y},=\operatorname{In}\left(\mathrm{y}+\sqrt{1}+\mathrm{y}^{2}\right), \cosh ^{-1}\) \(\mathrm{y}=\pm \operatorname{In}\left(\mathrm{y}+\sqrt{\mathrm{y}}^{2}-1\right)\), and \(\tanh ^{-1} \mathrm{y}=[1 / 2] \operatorname{In}[1+\mathrm{y} / 1-\mathrm{y}]\) show that (a) \(D_{y}\left(\sinh ^{-1} y\right)=\left[(1) /\left(\sqrt{1}+y^{2}\right)\right]\) (b) \(D_{y}\left(\cosh ^{-1} y\right)=\pm\left[(1) /\left(\sqrt{y}^{2}-1\right)\right]\) (c) \(D_{y}\left(\tanh ^{-1} y\right)=\left[(1) /\left(1-y^{2}\right)\right]\).

If \(\mathrm{y}=\sinh \mathrm{x}=[1 / 2]\left(\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}\right)\), the inverse function is written \(\mathrm{x}=\sinh ^{-1} \mathrm{y}\). Similar notations are employed for the inverses of the remaining hyperbolic functions. Show that: (a) Tanh \(^{-1} \mathrm{y}=\operatorname{In}\left\\{\mathrm{y}+\sqrt{ \left.\left(1+\mathrm{y}^{2}\right)\right\\}}\right.\) (c) \(\operatorname{Cosh}^{-1} \mathrm{y}=+\) in \(\\{\mathrm{y}+\sqrt{(\mathrm{y} 2-1)\\}}\) (d) Tanh \(^{-1} \mathrm{y}=[1 / 2] \operatorname{In}[(1+\mathrm{y}) /(1-\mathrm{y})]\).