derivative-x^{-2}e^{1/x}

解

導関数

- x^{- 2} e^{\frac{1}{x}}

\frac{2 e ^{\frac{1}{x}}}{x ^{3}} + \frac{e ^{\frac{1}{x}}}{x ^{4}}

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{- 2}, g = e^{\frac{1}{x}}

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

\frac{d}{d x} (x^{- 2}) = - \frac{2}{x ^{3}}

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

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

簡素化

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

= \frac{2 e ^{\frac{1}{x}}}{x ^{3}} + \frac{e ^{\frac{1}{x}}}{x ^{4}}