逆双曲線関数
d d x arsinh x = 1 1 + x 2 d d x arcosh x = 1 x 2 − 1 d d x artanh x = 1 1 − x 2 d d x arcoth x = 1 1 − x 2 d d x arsech x = − 1 x ( x + 1 ) 1 − x 1 + x d d x arcsch x = − 1 x 2 1 + 1 x 2 {\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} \,x&{}={\frac {1}{\sqrt {1+x^{2}}}}\\{\frac {d}{dx}}\operatorname {arcosh} \,x&{}={\frac {1}{\sqrt {x^{2}-1}}}\\{\frac {d}{dx}}\operatorname {artanh} \,x&{}={\frac {1}{1-x^{2}}}\\{\frac {d}{dx}}\operatorname {arcoth} \,x&{}={\frac {1}{1-x^{2}}}\\{\frac {d}{dx}}\operatorname {arsech} \,x&{}={\frac {-1}{x(x+1)\,{\sqrt {\frac {1-x}{1+x}}}}}\\{\frac {d}{dx}}\operatorname {arcsch} \,x&{}={\frac {-1}{x^{2}\,{\sqrt {1+{\frac {1}{x^{2}}}}}}}\\\end{aligned}}}