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# Math Formulas: Hyperbolic functions

### Definitions of hyperbolic functions

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

### Derivatives

 $$\frac{d}{dx}\, \sinh x = \cosh x$$
 $$\frac{d}{dx}\, \cosh x = \sinh x$$
 $$\frac{d}{dx}\, \tanh x = \mathrm{sech}^2x$$
 $$\frac{d}{dx}\, \mathrm{csch}\,x = -\mathrm{csch}\,x\cdot \coth x$$
 $$\frac{d}{dx}\, \mathrm{sech}\,x = -\mathrm{sech}\,x\cdot \tanh x$$
 $$\frac{d}{dx}\,\coth x = -\mathrm{csch}^2x$$

### Hyperbolic identities

 $$\cosh^2x - \sinh^2x = 1$$
 $$\tanh^2x + \mathrm{sech}^2x = 1$$
 $$\coth^2x - \mathrm{csch}^2x = 1$$
 $$\sinh(x \pm y) = \sinh x \cdot \cosh y \pm \cosh x\cdot \sinh y$$
 $$\cosh(x \pm y) = \cosh x \cdot \cosh y \pm \sinh x \cdot \sinh y$$
 $$\sinh(2\cdot x) = 2 \cdot \sinh x \cdot \cosh x$$
 $$\cosh(2\cdot x) = \cosh^2x + \sinh^2x$$
 $$\sinh^2x = \frac{-1 + \cosh 2x}{2}$$
 $$\cosh^2x = \frac{1 + \cosh 2x}{2}$$

### Inverse Hyperbolic functions

 $$\sinh^{-1}x=\ln \left(x+\sqrt{x^2 + 1}\right), ~~ x \in (-\infty, \infty)$$
 $$\cosh^{-1}x=\ln\left(x+\sqrt{x^2 - 1}\right), ~~ x \in [1, \infty)$$
 $$\tanh^{-1}x=\frac{1}{2} \ln\left(\frac{1 + x}{1 -x}\right), ~~ x \in (-1, 1)$$
 $$\coth^{-1}x=\frac{1}{2}\,\ln\left(\frac{x + 1}{x-1}\right), ~~ x \in (-\infty, -1) \cup (1, \infty)$$
 $$\mathrm{sech}^{-1}x=\ln\left(\frac{1 + \sqrt{1-x^2}}{x}\right), ~~ x \in (0, 1]$$
 $$\mathrm{csch}^{-1}x = \ln\left(\frac{1}{x} + \frac{\sqrt{1-x^2}}{|x|}\right), ~~ x \in (-\infty, 0) \cup (0,\infty)$$

### Derivatives of Inverse Hyperbolic functions

 $$\frac{d}{dx}\,\sinh^{-1}x= \frac{1}{\sqrt{x^2+1}}$$
 $$\frac{d}{dx}\, \cosh^{-1}x=\frac{1}{\sqrt{x^2-1}}$$
 $$\frac{d}{dx}\,tanh^{-1}x=\frac{1}{1-x^2}$$
 $$\frac{d}{dx}\, \mathrm{csch}^{-1}x=-\frac{1}{|x|\sqrt{1 + x^2}}$$
 $$\frac{d}{dx}\,\mathrm{sech}^{-1}x=-\frac{1}{x\sqrt{1 - x^2}}$$
 $$\frac{d}{dx}\,\coth^{-1}x=\frac{1}{1-x^2}$$

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