derivative y= x/((x+1)^{1/2)}

解

導関数

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

\frac{x + 2}{2 ( x + 1 ) ^{\frac{3}{2}}}

解答ステップ

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

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

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

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

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

簡素化

\frac{1 \cdot ( x + 1 ) ^{\frac{1}{2}} - \frac{1}{2 ( x + 1 ) ^{\frac{1}{2}}} x}{( ( x + 1 ) ^{\frac{1}{2}} ) ^{2}} : \frac{x + 2}{2 ( x + 1 ) ^{\frac{3}{2}}}

= \frac{x + 2}{2 ( x + 1 ) ^{\frac{3}{2}}}