Name        

Notation        

Equivalence

Derivative

Special properties

Hyperbolic Sine

sinh(x)

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

$\frac{d}{\mathrm{dx}}\sinh \left(x\right) \= \cosh \left(x\right)$

$\sinh \left(0\right) \= 0$

Hyperbolic Cosine

cosh(x)

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

$\frac{d}{\mathrm{dx}}\cosh \left(x\right) \= \sinh \left(x\right)$

$\cosh \left(0\right) \= 1$

Hyperbolic Tangent

tanh(x)

$\frac{\sinh \left(x\right)}{\cosh \left(x\right)}\= \frac{e^x - e^{-x}}{e^x \+ e^{-x}} \= \frac{e^{2x}-1}{e^{2x} \+ 1}$

$\frac{d}{\mathrm{dx}}\tanh \left(x\right) \= {\mathrm{sech}}^2{\left(x\right)}^{}$

$\tanh \left(0\right) \= 0$

Hyperbolic Cosecant

csch(x)

${\left(\sinh \left(x\right)\right)}^{-1} \= \frac{2}{e^x - e^{-x}}$

$\frac{d}{\mathrm{dx}}\mathrm{csch}\left(x\right) \= -\coth \left(x\right)\mathrm{csch}\left(x\right)$

Hyperbolic Secant

sech(x)

${\left(\cosh \left(x\right)\right)}^{-1} \= \frac{2}{e^x \+ e^{-x}}$

$\frac{d}{\mathrm{dx}}\mathrm{sech}\left(x\right) \= -\mathrm{sech}\left(x\right)\tanh \left(x\right)$

$\mathrm{sech}\left(0\right) \= 1$

Hyperbolic Cotangent

coth(x)

$\frac{\cosh \left(x\right)}{\sinh \left(x\right)}\= \frac{e^x \+ e^{-x}}{e^x - e^{-x}} \= \frac{e^{2x}\+1}{e^{2x} - 1}$

$\frac{d}{\mathrm{dx}}\coth \left(x\right) \= -{\mathrm{csch}}^2\left(x\right)$

$\coth \left(\frac{x}{2}\right)-\coth \left(x\right) \= \mathrm{csch}\left(x\right)$