Prove the formulas given in this table for the derivatives of the functions cosh, tanh,...

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Advance Math

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)

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Proving the formula fracdd xoperatornamecoth xoperatornamecsch2 x eginaligned fracdd xoperatornamecoth xfracdd xleftfraccosh xsinh x ight Rightarrow fracdd xoperatornamecoth xleftfracleftsinh x fracdd x cosh xcosh x fracdd x sinh x ightsinh 2 x ight quad ext By quotient rule of differentiation Rightarrow fracdd xoperatornamecoth xleftfracsinh x cdot sinh xcosh x cdot cosh xsinh 2 x ight quadleftoperatornamesince fracdd x sinh xcosh x fracdd x cosh xsinh x ight Rightarrow fracdd xoperatornamecoth xleftfracsinh 2 xcosh 2 xsinh 2 x ight Rightarrow fracdd xoperatornamecoth xleftfrac1sinh 2 x See Answer

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