derivative of (xln(x)/(e^x))

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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