Hyperbolic Trigonometric Functions

$\cosh x$

$\cosh x=\frac{e^x+e^{-x}}{2}$ ${\cosh }^{\prime }x=\sinh x$ ${\cosh }^{-1}x=\ln (x+\sqrt{x^2-1})$ $({\cosh }^{-1}{)}^{\prime }x=\frac{1}{\sqrt{x^2-1}}$ These are used to model Catenary shapes

Range

$\cosh \ge 1$

$\sinh x$

$\sinh x=\frac{e^x-e^{-x}}{2}$ ${\sinh }^{\prime }x=\cosh x$ ${\sinh }^{-1}(x)=\ln (x+\sqrt{x^2+1})$ $({\sinh }^{-1}{)}^{\prime }x=\frac{1}{\sqrt{x^2+1}}$

$\tanh x$

$\tanh x=\frac{\sinh x}{\cosh x}$ ${\tanh }^{\prime }x=sec{h}^2(x)$ ${\tanh }^{-1}=\frac{1}{2}\ln (\frac{1+x}{1-x})$ $({\tanh }^{-1}{)}^{\prime }x=\frac{1}{1-x^2}$

$cschx$

$cschx=\frac{1}{\sinh x}$

$sechx$

$sechx=\frac{1}{\cosh x}$

$cothx$

$cothx=\frac{\cosh x}{\sinh x}$