derivative of (xsin(x)/(1+x))

解

\frac{d}{d x} (\frac{x sin ( x )}{1 + x})

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

解答ステップ

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

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

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

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

簡素化

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

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