Prove the formulas given in this table for the derivatives of the functions cosh, tanh, csch, sech, and coth. Which of the following are proven correctly? (Select all that apply.)
(square frac{d}{d x}(operatorname{coth} x)=frac{d}{d x}left(frac{sinh x}{cosh x}ight)=frac{cosh x cosh x-sinh x sinh x}{cosh ^{2} x}=frac{cosh ^{2} x-sinh ^{2} x}{cosh ^{2} x}=-frac{1}{cosh ^{2} x}=-operatorname{csch}^{2} x) (square frac{d}{d x}(operatorname{csch} x)=frac{d}{d x}left(frac{1}{sinh x}ight)=-frac{cosh x}{sinh ^{2} x}=-frac{1}{sinh x} cdot frac{cosh x}{sinh x}=-operatorname{csch} x operatorname{coth} x)
(square frac{d}{d x}(cosh x)=frac{d}{d x}left[frac{1}{2}left(e^{x}-e^{-x}ight)ight]=frac{1}{2}left(e^{x}+e^{-x}ight)=sinh x)
(square frac{d}{d x}(operatorname{csch} x)=frac{d}{d x}left(frac{1}{sinh x}ight)=-frac{cosh ^{2} x}{sinh ^{2} x}=-frac{1}{sinh x} cdot frac{cosh ^{2} x}{sinh x}=-operatorname{csch} x operatorname{coth} x)
(square frac{d}{d x}(operatorname{sech} x)=frac{d}{d x}left(frac{1}{cosh x}ight)=-frac{sinh x}{cosh ^{2} x}=-frac{1}{cosh x} cdot frac{sinh x}{cosh x}=-operatorname{sech} x anh x)
