(e^{1/x}-(e^{1/x})/x-(5e^{1/x})/(x^2))^'

解

(e^{\frac{1}{x}} - \frac{e ^{\frac{1}{x}}}{x} - \frac{5 e ^{\frac{1}{x}}}{x ^{2}})^{^{'}}

\frac{11 e ^{\frac{1}{x}} x + 5 e ^{\frac{1}{x}}}{x ^{4}}

解答ステップ

(e^{\frac{1}{x}} - \frac{e ^{\frac{1}{x}}}{x} - \frac{5 e ^{\frac{1}{x}}}{x ^{2}})^{'}

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= (e^{\frac{1}{x}})^{^{'}} - (\frac{e ^{\frac{1}{x}}}{x})^{^{'}} - (\frac{5 e ^{\frac{1}{x}}}{x ^{2}})^{^{'}}

(e^{\frac{1}{x}})^{'} = - \frac{e ^{\frac{1}{x}}}{x ^{2}}

(\frac{e ^{\frac{1}{x}}}{x})^{'} = \frac{- e ^{\frac{1}{x}} - e ^{\frac{1}{x}} x}{x ^{3}}

(\frac{5 e ^{\frac{1}{x}}}{x ^{2}})^{'} = \frac{5 ( - e ^{\frac{1}{x}} - 2 e ^{\frac{1}{x}} x )}{x ^{4}}

= - \frac{e ^{\frac{1}{x}}}{x ^{2}} - \frac{- e ^{\frac{1}{x}} - e ^{\frac{1}{x}} x}{x ^{3}} - \frac{5 ( - e ^{\frac{1}{x}} - 2 e ^{\frac{1}{x}} x )}{x ^{4}}

簡素化

- \frac{e ^{\frac{1}{x}}}{x ^{2}} - \frac{- e ^{\frac{1}{x}} - e ^{\frac{1}{x}} x}{x ^{3}} - \frac{5 ( - e ^{\frac{1}{x}} - 2 e ^{\frac{1}{x}} x )}{x ^{4}} : \frac{11 e ^{\frac{1}{x}} x + 5 e ^{\frac{1}{x}}}{x ^{4}}

= \frac{11 e ^{\frac{1}{x}} x + 5 e ^{\frac{1}{x}}}{x ^{4}}