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Hyperbolic Trigonometry

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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

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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$\frac{du}{dx}$
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]

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