双曲線関数
}}=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\dotsb ,\quad |x|<{\frac {\pi }{2}}\\[1ex]\operatorname {csch} x&={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}}={\frac {1}{x}}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\dotsb ,\quad 0<|x|<\pi \\[1ex]\operatorname {sech} x&=\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}}=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\dotsb ,\quad |x|<{\frac {\pi }{2}}\\[1ex]\coth x&={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}}={\frac {1}{x}}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\dotsb ,\quad 0<|x|<\pi \end{aligned}}}
d d x sinh  x = cosh  x d d x cosh  x = sinh  x d d x tanh  x = 1 − tanh 2  x = sech 2  x = 1 cosh 2  x d d x coth  x = 1 − coth 2  x = − csch 2  x = − 1 sinh 2  x d d x csch  x = − coth  x csch  x d d x sech  x = − tanh  x sech  x {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\sinh x&=\cosh x\\{\frac {\mathrm {d} }{\mathrm {d} x}}\cosh x&=\sinh x\\{\frac {\mathrm {d} }{\mathrm {d} x}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {csch} x&=-\coth x\operatorname {csch} x\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\end{aligned}}}