derivative of (2x/(2+sqrt(x)))

解

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

\frac{4 + x}{( 2 + x ) ^{2}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

2 \cdot \frac{1 \cdot ( 2 + x ) - \frac{1}{2 x} x}{( 2 + x ) ^{2}} : \frac{4 + x}{( 2 + x ) ^{2}}

= \frac{4 + x}{( 2 + x ) ^{2}}