derivative of ln(xe^x)

解

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

\frac{e ^{x}}{x} + e^{x} ln (x)

解答ステップ

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

積の法則を適用:

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

f = ln (x), g = e^{x}

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

\frac{d}{d x} (ln (x)) = \frac{1}{x}

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

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

\frac{1}{x} e^{x} = \frac{e ^{x}}{x}

= \frac{e ^{x}}{x} + e^{x} ln (x)