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Question Number 134204 by liberty last updated on 01/Mar/21

$$\mathrm{f}\left(\mathrm{x}\right)=\sqrt{2(1-\cos 2\mathrm{x})}.$$$$\mathrm{find}\mathrm{f}{}^{\prime }\left(0\right).$$

Answered by bramlexs22 last updated on 02/Mar/21

$$\mathrm{f}\left(\mathrm{x}\right)=\sqrt{2\left(2\sin {}^2\mathrm{x}\right)}=\mid 4\sin \mathrm{x}\mid =2\mid \sin \mathrm{x}\mid$$$$\mathrm{f}{}^{\prime }\left(0^{-}\right)=\underset{x\to 0^{-}}{\lim }\frac{-2\sin \mathrm{x}-0}{\mathrm{x}-0}=-2$$$$\mathrm{f}{}^{\prime }\left(0^{+}\right)=\underset{x\to 0^{+}}{\lim }\frac{2\sin \mathrm{x}-0}{\mathrm{x}-0}=2$$$$\mathrm{then}\mathrm{f}{}^{\prime }\left(0^{-}\right)\ne \mathrm{f}{}^{\prime }\left(0^{+}\right)\mathrm{so}\mathrm{f}{}^{\prime }\left(0\right)\mathrm{is}$$$${\mathrm{doesn}}^{\prime }\mathrm{t}\mathrm{exist}$$

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