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# Hyperbolic Trigonometry

Top
 Sub Topics Hyperbolic functions are similar to the trigonometric functions and they are defined in terms of the exponential function. The trigonometric functions expressed in the form of e$^{x}$ are known as hyperbolic trigonometric functions.Six basic hyperbolic functions are given below: sinh x = $\frac{e^{x} - e^{-x}}{2}$ cosh x =  $\frac{e^{x} + e^{-x}}{2}$tanh x = $\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}$coth x = $\frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}$sech x =  $\frac{2}{e^{x} + e^{-x}}$csch x =  $\frac{2}{e^{x} - e^{-x}}$

## Derivative of Hyperbolic Functions

There are six hyperbolic functions and they are defined as follows:

$\frac{d}{dx}$ sinh u = cosh u$\frac{du}{dx}$

$\frac{d}{dx}$ cosh u = sinh u$\frac{du}{dx}$

$\frac{d}{dx}$ tanh u = sech$^{2}$ u$\frac{du}{dx}$

$\frac{d}{dx}$ coth u = -csch$^{2}$ u$\frac{du}{dx}$

$\frac{d}{dx}$ sech u = -sech u tanh u$\frac{du}{dx}$

$\frac{d}{dx}$ csch u = -csch u coth u$\frac{du}{dx}$

## Derivatives of Inverse Hyperbolic Functions

Formula for derivatives of six hyperbolic functions are given below:

$\frac{d}{dx}$ $\sinh ^{-1} u(x)$ = $\frac{1}{\sqrt{u(x)^{2}+1}}\ \frac{\mathrm{d}u(x)}{\mathrm{d} x}$

$\frac{d}{dx}$ $\cosh ^{-1} u(x)$ = $\pm$$\frac{1}{\sqrt{u(x)^{2}-1}}\ \frac{\mathrm{d}u(x)}{\mathrm{d} x}$
where, [ +ve, if cosh$^{-1}$u(x) > 0, u(x)>1] and [ -ve, if cosh$^{-1}$u(x) < 0, u(x)>1 ]

$\frac{d}{dx}$ $\tanh ^{-1} u(x)$ = $\frac{1}{1 - u^{2}(x)} \frac{du(x)}{dx}$, [ -1 < u(x) < 1]

$\frac{d}{dx}$ sech $^{-1} u(x)$ = $\frac{\pm1}{u(x)\sqrt{1-u^{2}(x)}}$ $\frac{du(x)}{dx}$
where, [+ve, if sech$^{-1}$u(x) < 0, 0<u(x)<1] and [-ve, if sech$^{-1}$u(x) > 0, 0<u(x)<1 ]

$\frac{d}{dx}$ csch $^{-1} u(x)$ = $\frac{-1}{|u(x)|\sqrt{1+u^{2}(x)}}.\frac{\pm1}{u(x)\sqrt{1+u^{2}(x)}}\frac{du(x)}{dx}$
[-ve, if u(x) > 0, +ve, if u(x) < 0] and [u(x) > 1, or u(x) , -1]

$\frac{d}{dx}$ $\coth ^{-1} u(x)$ = $\frac{1}{1 - u^{2}(x)} \frac{du(x)}{dx}$