derivative of (y-sqrt(x)^{1/2})

解

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

\frac{2 x ^{\frac{1}{2}} \frac{d y}{d x} - 1}{4 x ^{\frac{1}{2}} ( y - x ) ^{\frac{1}{2}}}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

\frac{1}{2 ( y - x ) ^{\frac{1}{2}}} (\frac{d y}{d x} - \frac{1}{2 x ^{\frac{1}{2}}}) : \frac{2 x ^{\frac{1}{2}} \frac{d y}{d x} - 1}{4 x ^{\frac{1}{2}} ( y - x ) ^{\frac{1}{2}}}

= \frac{2 x ^{\frac{1}{2}} \frac{d y}{d x} - 1}{4 x ^{\frac{1}{2}} ( y - x ) ^{\frac{1}{2}}}