derivative of (xcos(x)/(1+e^x))

解

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

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

解答ステップ

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

\frac{d}{d x} (x cos (x)) = cos (x) - x sin (x)

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

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