The derivative of a hyperbolic function with respect to a variable (used as real number of the function) is called differentiation of the hyperbolic function with respect to that variable.

Take $x$ and assume it represents a real number. The six hyperbolic functions are written as $\sinh{x}$, $\cosh{x}$, $\tanh{x}$, $\coth{x}$, $\operatorname{sech}{x}$ and $\operatorname{csch}{x}$ or $\operatorname{cosech}{x}$. The derivative of each hyperbolic function with respect to $x$ is listed here for beginners.

01

$\large \dfrac{d}{dx} \, \sinh{x} \,=\, \cosh{x}$

02

$\large \dfrac{d}{dx} \, \cosh{x} \,=\, \sinh{x}$

03

$\large \dfrac{d}{dx} \, \tanh x = \operatorname{sech}^2{x}$

04

$\large \dfrac{d}{dx} \, \coth x = -\operatorname{csch}^2{x}$

05

$\large \dfrac{d}{dx} \, \operatorname{sech} x = -\operatorname{sech}{x}\tanh{x}$

06

$\large \dfrac{d}{dx} \, \operatorname{csch} x = -\operatorname{csch}{x}\coth{x}$

The six differentiation rules of hyperbolic functions are used as formulas in differential calculus mathematics.

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