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Найти производную функции \(y=\frac{2}{3}\sqrt{(arctg(e^{x}))^{3}} \). Производную буде искать по формуле производной сложной функции $$y'=(\frac{2}{3}\sqrt{(arctg(e^{x}))^{3}})' = \frac{2}{3}((arctg(e^{x}))^{\frac{3}{2}})' =\\ \frac{2}{3}*\frac{3}{2}*(arctg(e^{x}))^{\frac{3}{2}-1}*\frac{1}{1 + (e^x)^2}*e^x= \sqrt{arctg(e^{x})}*\frac{e^x}{1 + e^{2x}}$$Ответ: производная функции \( (=\frac{2}{3}\sqrt{(arctge^{x})^{3}})' = \sqrt{arctg(e^{x})}*\frac{e^x}{1 + e^{2x}}\)