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Question Number 51436 by Tawa1 last updated on 26/Dec/18

$$\mathrm{If}\mathrm{y}=\frac{({\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}}).\tanh \mathrm{x}}{{\mathrm{e}}^{\mathrm{x}}-\sinh \mathrm{x}}$$$$\mathrm{prove}\mathrm{that}{\mathrm{y}}^{\prime }=2{\mathrm{sech}}^2\mathrm{x}$$

Answered by peter frank last updated on 26/Dec/18

$$y(e^x-\sinh x)=(e^x+e^{-x})\tanh x$$$$recall$$$$\sinh x=\frac{e^x-e^{-x}}{2}$$$$\tanh x=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$$$substituteandsimplify$$$$y=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$$$y^{{}^{\prime }}=\frac{2.2}{{(e^x+e^{-x})}^2}$$$$y^{{}^{\prime }}=\frac{4}{{(e^x+e^{-x})}^2}=\mathrm{sech}{}^{2}x$$$$y^{{}^{\prime }}=\mathrm{sech}{}^{2}x$$$$plzcheck$$$$$$

Commented by Tawa1 last updated on 27/Dec/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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