derivative of (2e^x/(5x))

解

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

\frac{d x}{d x} = 1

= \frac{2}{5} \cdot \frac{e ^{x} x - 1 \cdot e ^{x}}{x ^{2}}

簡素化

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

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