d d x sinh ⁡ x = cosh ⁡ x {\displaystyle {\frac {d}{dx}}\sinh x=\cosh x\,} d d x cosh ⁡ x = sinh ⁡ x {\displaystyle {\frac {d}{dx}}\cosh x=\sinh x\,} d d x tanh ⁡ x = 1 − tanh 2 ⁡ x = sech 2 x = 1 / cosh 2 ⁡ x {\displaystyle {\frac {d}{dx}}\tanh x=1-\tanh ^{2}x={\hbox{sech}}^{2}x=1/\cosh ^{2}x\,} d d x coth ⁡ x = 1 − coth 2 ⁡ x = − csch 2 x = − 1 / sinh 2 ⁡ x {\displaystyle {\frac {d}{dx}}\coth x=1-\coth ^{2}x=-{\hbox{csch}}^{2}x=-1/\sinh ^{2}x\,} d d x   csch x = − coth ⁡ x   csch x {\displaystyle {\frac {d}{dx}}\ {\hbox{csch}}\,x=-\coth x\ {\hbox{csch}}\,x\,} d d x   sech x = − tanh ⁡ x   sech x {\displaystyle {\frac {d}{dx}}\ {\hbox{sech}}\,x=-\tanh x\ {\hbox{sech}}\,x\,} d d x arsinh x = 1 x 2 + 1 {\displaystyle {\frac {d}{dx}}\,\operatorname {arsinh} \,x={\frac {1}{\sqrt {x^{2}+1}}}} d d x arcosh x = 1 x 2 − 1 {\displaystyle {\frac {d}{dx}}\,\operatorname {arcosh} \,x={\frac {1}{\sqrt {x^{2}-1}}}} d d x artanh x = 1 1 − x 2 , | x | < 1 {\displaystyle {\frac {d}{dx}}\,\operatorname {artanh} \,x={\frac {1}{1-x^{2}}},\left|x\right|<1} d d x arcoth x = 1 1 − x 2 , | x | > 1 {\displaystyle {\frac {d}{dx}}\,\operatorname {arcoth} \,x={\frac {1}{1-x^{2}}},\left|x\right|>1} d d x arsech x = − 1 x 1 − x 2 , 0 < x < 1 {\displaystyle {\frac {d}{dx}}\,\operatorname {arsech} \,x=-{\frac {1}{x{\sqrt {1-x^{2}}}}},0<x<1} d d x arcsch x = − 1 | x | 1 + x 2 , x ≠ 0 {\displaystyle {\frac {d}{dx}}\,\operatorname {arcsch} \,x=-{\frac {1}{\left|x\right|{\sqrt {1+x^{2}}}}},x\neq 0}