All Maths Formulas

Algebra

Functions TrigExpLogHyp

Analytic Geometry

Limits, Derivatives

Indefinite Integrals

Definite Integrals

Series Arith Geo Taylor

Hyperbolic functions

Definitions of hyperbolic functions

sinh x = e x - e - x 2

$\sinh x=\frac{e^x - e^{-x}}{2}$

cosh x = e x + e - x 2

$\cosh x=\frac{e^x + e^{-x}}{2}$

tanh x = e x - e - x e x + e - x = sinh x cosh x

$\tanh x=\frac{e^x - e^{-x}}{e^x + e^{-x}} =\frac{\sinh x}{\cosh x}$

c s c h x = 2 e x - e - x = 1 sinh x

$\mathrm{csch}\,x=\frac{2}{e^x - e^{-x}} = \frac{1}{\sinh x}$

s e c h x = 2 e x + e - x = 1 cosh x

$\mathrm{sech}\,x=\frac{2}{e^x + e^{-x}} = \frac{1}{\cosh x}$

coth x = e x + e - x e x - e - x = cosh x sinh x

$\coth\,x=\frac{e^x + e^{-x}}{e^x - e^{-x}} = \frac{\cosh x}{\sinh x}$

Derivatives

d d x sinh x = cosh x

$\frac{d}{dx}\, \sinh x = \cosh x$

d d x cosh x = sinh x

$\frac{d}{dx}\, \cosh x = \sinh x$

d d x tanh x = s e c h 2 x

$\frac{d}{dx}\, \tanh x = \mathrm{sech}^2x$

d d x c s c h x = - c s c h x \cdot coth x

$\frac{d}{dx}\, \mathrm{csch}\,x = -\mathrm{csch}\,x\cdot \coth x$

d d x s e c h x = - s e c h x \cdot tanh x

$\frac{d}{dx}\, \mathrm{sech}\,x = -\mathrm{sech}\,x\cdot \tanh x$

d d x coth x = - c s c h 2 x

$\frac{d}{dx}\,\coth x = -\mathrm{csch}^2x$

Hyperbolic identities

cosh 2 x - sinh 2 x = 1

$\cosh^2x - \sinh^2x = 1$

tanh 2 x + s e c h 2 x = 1

$\tanh^2x + \mathrm{sech}^2x = 1$

coth 2 x - c s c h 2 x = 1

$\coth^2x - \mathrm{csch}^2x = 1$

sinh (x \pm y) = sinh x \cdot cosh y \pm cosh x \cdot sinh y

$\sinh(x \pm y) = \sinh x \cdot \cosh y \pm \cosh x\cdot \sinh y$

cosh (x \pm y) = cosh x \cdot cosh y \pm sinh x \cdot sinh y

$\cosh(x \pm y) = \cosh x \cdot \cosh y \pm \sinh x \cdot \sinh y$

sinh (2 \cdot x) = 2 \cdot sinh x \cdot cosh x

$\sinh(2\cdot x) = 2 \cdot \sinh x \cdot \cosh x$

cosh (2 \cdot x) = cosh 2 x + sinh 2 x

$\cosh(2\cdot x) = \cosh^2x + \sinh^2x$

sinh 2 x = - 1 + cosh 2 x 2

$\sinh^2x = \frac{-1 + \cosh 2x}{2}$

cosh 2 x = 1 + cosh 2 x 2

$\cosh^2x = \frac{1 + \cosh 2x}{2}$

Inverse Hyperbolic functions

sinh - 1 x = ln (x + x 2 + 1 - - - - - \sqrt), x \in (- \infty, \infty)

$\sinh^{-1}x=\ln \left(x+\sqrt{x^2 + 1}\right), ~~ x \in (-\infty, \infty)$

cosh - 1 x = ln (x + x 2 - 1 - - - - - \sqrt), x \in [1, \infty)

$\cosh^{-1}x=\ln\left(x+\sqrt{x^2 - 1}\right), ~~ x \in [1, \infty)$

tanh - 1 x = 1 2 ln (1 + x 1 - x), x \in (- 1, 1)

$\tanh^{-1}x=\frac{1}{2} \ln\left(\frac{1 + x}{1 -x}\right), ~~ x \in (-1, 1)$

coth - 1 x = 1 2 ln (x + 1 x - 1), x \in (- \infty, - 1) \cup (1, \infty)

$\coth^{-1}x=\frac{1}{2}\,\ln\left(\frac{x + 1}{x-1}\right), ~~ x \in (-\infty, -1) \cup (1, \infty)$

s e c h - 1 x = ln (1 + 1 - x 2 - - - - - \sqrt x), x \in (0, 1]

$\mathrm{sech}^{-1}x=\ln\left(\frac{1 + \sqrt{1-x^2}}{x}\right), ~~ x \in (0, 1]$

c s c h - 1 x = ln (1 x + 1 - x 2 - - - - - \sqrt | x |), x \in (- \infty, 0) \cup (0, \infty)

$\mathrm{csch}^{-1}x = \ln\left(\frac{1}{x} + \frac{\sqrt{1-x^2}}{|x|}\right), ~~ x \in (-\infty, 0) \cup (0,\infty)$

Derivatives of Inverse Hyperbolic functions

d d x sinh - 1 x = 1 x 2 + 1 - - - - - \sqrt

$\frac{d}{dx}\,\sinh^{-1}x= \frac{1}{\sqrt{x^2+1}}$

d d x cosh - 1 x = 1 x 2 - 1 - - - - - \sqrt

$\frac{d}{dx}\, \cosh^{-1}x=\frac{1}{\sqrt{x^2-1}}$

d d x t a n h - 1 x = 1 1 - x 2

$\frac{d}{dx}\,tanh^{-1}x=\frac{1}{1-x^2}$

d d x c s c h - 1 x = - 1 | x | 1 + x 2 - - - - - \sqrt

$\frac{d}{dx}\, \mathrm{csch}^{-1}x=-\frac{1}{|x|\sqrt{1 + x^2}}$

d d x s e c h - 1 x = - 1 x 1 - x 2 - - - - - \sqrt

$\frac{d}{dx}\,\mathrm{sech}^{-1}x=-\frac{1}{x\sqrt{1 - x^2}}$

d d x coth - 1 x = 1 1 - x 2

$\frac{d}{dx}\,\coth^{-1}x=\frac{1}{1-x^2}$

