derivative (x^{1/2}-1)/(x^{1/2)}

解

導関数

\frac{x ^{\frac{1}{2}} - 1}{x ^{\frac{1}{2}}}

\frac{1}{2 x ^{\frac{3}{2}}}

解答ステップ

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

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

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

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

簡素化

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

= \frac{1}{2 x ^{\frac{3}{2}}}