逆双曲線関数
実数xに対して、
d d x arsech x = ∓ 1 x 1 − x 2 ; R { x } ≷ 0 d d x arcsch x = ∓ 1 x 1 + x 2 ; R { x } ≷ 0 {\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arsech} \,x&{}={\frac {\mp 1}{x\,{\sqrt {1-x^{2}}}}};\qquad \Re \{x\}\gtrless 0\\{\frac {d}{dx}}\operatorname {arcsch} \,x&{}={\frac {\mp 1}{x\,{\sqrt {1+x^{2}}}}};\qquad \Re \{x\}\gtrless 0\end{aligned}}}
微分法の例:θ = arsinh xとおくと、
d arsinh x d x = d θ d sinh  θ = 1 cosh  θ = 1 1 + sinh 2  θ = 1 1 + x 2 {\displaystyle {\frac {d\,\operatorname {arsinh} \,x}{dx}}={\frac {d\theta }{d\sinh \theta }}={\frac {1}{\cosh \theta }}={\frac {1}{\sqrt {1+\sinh ^{2}\theta }}}={\frac {1}{\sqrt {1+x^{2}}}}}