derivative of (e^x/(6+e^x))

解

\frac{d}{d x} (\frac{e ^{x}}{6 + e ^{x}})

\frac{6 e ^{x}}{( 6 + e ^{x} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{e ^{x}}{6 + 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} ( e ^{x} ) ( 6 + e ^{x} ) - \frac{d}{d x} ( 6 + e ^{x} ) e ^{x}}{( 6 + e ^{x} ) ^{2}}

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

\frac{d}{d x} (6 + e^{x}) = e^{x}

= \frac{e ^{x} ( 6 + e ^{x} ) - e ^{x} e ^{x}}{( 6 + e ^{x} ) ^{2}}

簡素化

\frac{e ^{x} ( 6 + e ^{x} ) - e ^{x} e ^{x}}{( 6 + e ^{x} ) ^{2}} : \frac{6 e ^{x}}{( 6 + e ^{x} ) ^{2}}

= \frac{6 e ^{x}}{( 6 + e ^{x} ) ^{2}}