{\ displaystyle{\开始{对齐}{\压裂{d} {dx}} \ operatorname {arsinh}({} ={\压裂{1}{\√6 {x ^{2} + 1}}},{\文本对所有真正}{}x \ \{\压裂{d} {dx}} \ operatorname {arcosh}({} ={\压裂{1}{\√6 x ^{1}{2}}},{\文本对所有真正}{}x > 1 \ \{\压裂{d} {dx}} \ operatorname {artanh}({} ={\压裂{1}{1 - x ^{2}}},{\文本对所有真正}{}| | x < 1 \ \{\压裂{d} {dx}} \ operatorname {arcoth}({} ={\压裂{1}{1 - x ^{2}}},{\文本对所有真正}{}| x | > 1 \ \{\压裂{d} {dx}} \ operatorname {arsech}({} ={\压裂{1}{x{\√6 {1 - x ^{2}}}}},{{\文本for all real }}x\in (0,1)\\{\frac {d}{dx}}\operatorname {arcsch} x&{}={\frac {-1}{|x|{\sqrt {1+x^{2}}}}},{\text{ for all real }}x{\text{, except }}0\\\end{aligned}}}