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Math formulas: Hyperbolic functions

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Definitions of hyperbolic functions

sinhx=exex2 \sinh x=\frac{e^x - e^{-x}}{2}
coshx=ex+ex2 \cosh x=\frac{e^x + e^{-x}}{2}
tanhx=exexex+ex=sinhxcoshx \tanh x=\frac{e^x - e^{-x}}{e^x + e^{-x}} =\frac{\sinh x}{\cosh x}
cschx=2exex=1sinhx \mathrm{csch}\,x=\frac{2}{e^x - e^{-x}} = \frac{1}{\sinh x}
sechx=2ex+ex=1coshx \mathrm{sech}\,x=\frac{2}{e^x + e^{-x}} = \frac{1}{\cosh x}
cothx=ex+exexex=coshxsinhx \coth\,x=\frac{e^x + e^{-x}}{e^x - e^{-x}} = \frac{\cosh x}{\sinh x}

Derivatives

ddxsinhx=coshx \frac{d}{dx}\, \sinh x = \cosh x
ddxcoshx=sinhx \frac{d}{dx}\, \cosh x = \sinh x
ddxtanhx=sech2x \frac{d}{dx}\, \tanh x = \mathrm{sech}^2x
ddxcschx=cschxcothx \frac{d}{dx}\, \mathrm{csch}\,x = -\mathrm{csch}\,x\cdot \coth x
ddxsechx=sechxtanhx \frac{d}{dx}\, \mathrm{sech}\,x = -\mathrm{sech}\,x\cdot \tanh x
ddxcothx=csch2x \frac{d}{dx}\,\coth x = -\mathrm{csch}^2x

Hyperbolic identities

cosh2xsinh2x=1 \cosh^2x - \sinh^2x = 1
tanh2x+sech2x=1 \tanh^2x + \mathrm{sech}^2x = 1
coth2xcsch2x=1 \coth^2x - \mathrm{csch}^2x = 1
sinh(x±y)=sinhxcoshy±coshxsinhy \sinh(x \pm y) = \sinh x \cdot \cosh y \pm \cosh x\cdot \sinh y
cosh(x±y)=coshxcoshy±sinhxsinhy \cosh(x \pm y) = \cosh x \cdot \cosh y \pm \sinh x \cdot \sinh y
sinh(2x)=2sinhxcoshx \sinh(2\cdot x) = 2 \cdot \sinh x \cdot \cosh x
cosh(2x)=cosh2x+sinh2x \cosh(2\cdot x) = \cosh^2x + \sinh^2x
sinh2x=1+cosh2x2 \sinh^2x = \frac{-1 + \cosh 2x}{2}
cosh2x=1+cosh2x2 \cosh^2x = \frac{1 + \cosh 2x}{2}

Inverse Hyperbolic functions

sinh1x=ln(x+x2+1),  x(,) \sinh^{-1}x=\ln \left(x+\sqrt{x^2 + 1}\right), ~~ x \in (-\infty, \infty)
cosh1x=ln(x+x21),  x[1,) \cosh^{-1}x=\ln\left(x+\sqrt{x^2 - 1}\right), ~~ x \in [1, \infty)
tanh1x=12ln(1+x1x),  x(1,1) \tanh^{-1}x=\frac{1}{2} \ln\left(\frac{1 + x}{1 -x}\right), ~~ x \in (-1, 1)
coth1x=12ln(x+1x1),  x(,1)(1,) \coth^{-1}x=\frac{1}{2}\,\ln\left(\frac{x + 1}{x-1}\right), ~~ x \in (-\infty, -1) \cup (1, \infty)
sech1x=ln(1+1x2x),  x(0,1] \mathrm{sech}^{-1}x=\ln\left(\frac{1 + \sqrt{1-x^2}}{x}\right), ~~ x \in (0, 1]
csch1x=ln(1x+1x2x),  x(,0)(0,) \mathrm{csch}^{-1}x = \ln\left(\frac{1}{x} + \frac{\sqrt{1-x^2}}{|x|}\right), ~~ x \in (-\infty, 0) \cup (0,\infty)

Derivatives of Inverse Hyperbolic functions

ddxsinh1x=1x2+1 \frac{d}{dx}\,\sinh^{-1}x= \frac{1}{\sqrt{x^2+1}}
ddxcosh1x=1x21 \frac{d}{dx}\, \cosh^{-1}x=\frac{1}{\sqrt{x^2-1}}
ddxtanh1x=11x2 \frac{d}{dx}\,tanh^{-1}x=\frac{1}{1-x^2}
ddxcsch1x=1x1+x2 \frac{d}{dx}\, \mathrm{csch}^{-1}x=-\frac{1}{|x|\sqrt{1 + x^2}}
ddxsech1x=1x1x2 \frac{d}{dx}\,\mathrm{sech}^{-1}x=-\frac{1}{x\sqrt{1 - x^2}}
ddxcoth1x=11x2 \frac{d}{dx}\,\coth^{-1}x=\frac{1}{1-x^2}

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