derivative of-(5e^{5/x}/(x^2))

解

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= - 5 \cdot \frac{\frac{d}{d x} ( e ^{\frac{5}{x}} ) x ^{2} - \frac{d}{d x} ( x ^{2} ) e ^{\frac{5}{x}}}{( x ^{2} ) ^{2}}

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

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

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

簡素化

- 5 \cdot \frac{( - \frac{5 e ^{\frac{5}{x}}}{x ^{2}} ) x ^{2} - 2 x e ^{\frac{5}{x}}}{( x ^{2} ) ^{2}} : - \frac{5 ( - 5 e ^{\frac{5}{x}} - 2 e ^{\frac{5}{x}} x )}{x ^{4}}

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