derivative x/(sqrt(x)+1)

解

導関数

\frac{x}{x + 1}

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

解答ステップ

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

商の法則を適用:

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

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

\frac{d x}{d x} = 1

\frac{d}{d x} (x + 1) = \frac{1}{2 x}

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

簡素化

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

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