derivative of (6e^x/((6+e^x)^2))

解

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

\frac{6 e ^{x} ( - e ^{x} + 6 )}{( 6 + e ^{x} ) ^{3}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

6 \cdot \frac{e ^{x} ( 6 + e ^{x} ) ^{2} - 2 ( 6 + e ^{x} ) e ^{x} e ^{x}}{( ( 6 + e ^{x} ) ^{2} ) ^{2}} : \frac{6 e ^{x} ( - e ^{x} + 6 )}{( 6 + e ^{x} ) ^{3}}

= \frac{6 e ^{x} ( - e ^{x} + 6 )}{( 6 + e ^{x} ) ^{3}}