第6章 6.13

(1)

$\cosh^2 x - \sinh^2 x = \left(\frac{e^x+e^{-x}}{2}\right)^2 - \left(\frac{e^x-e^{-x}}{2}\right)^2 = \frac{e^{2x}+2+e^{-2x}-e^{2x}+2-e^{-2x}}{4} = 1$

(2)

$(\sinh x)' = \left(\frac{e^x-e^{-x}}{2}\right)' = \frac{1}{2}(e^x+e^{-x}) = \cosh x$

(3)

$(\cosh x)' = \left(\frac{e^x+e^{-x}}{2}\right)' = \frac{1}{2}(e^x-e^{-x}) = \sinh x$

(4)

$(\tanh x)' = \left(\frac{\sinh x}{\cosh x}\right)' = \left(\frac{e^x-e^{-x}}{e^x+e^{-x}}\right)' = \frac{(e^x+e^{-x})(e^x+e^{-x})-(e^x-e^{-x})(e^x-e^{-x})}{(e^x+e^{-x})^2}$

$= \frac{e^{2x}+2+e^{-2x}-e^{2x}+2-e^{-2x}}{(e^x+e^{-x})^2} = \frac{1}{\left(\dfrac{e^x+e^{-x}}{2}\right)^2} = \frac{1}{\cosh^2 x}$