derivative of e^x 1/x

解

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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