There are six derivative rules to evaluate the differentiation of the hyperbolic functions in differential calculus. So, learn every derivative formula of hyperbolic functions with mathematical proofs.

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

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

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

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

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

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

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