derivative of arctan(sqrt((x-y/x)))

解

\frac{d}{d x} (arctan (\frac{x - y}{x}))

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

解答ステップ

\frac{d}{d x} (arctan (\frac{x - y}{x}))

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{x - y}{x}) = \frac{- x \frac{d y}{d x} + y}{2 x ^{\frac{3}{2}} x - y}

= \frac{1}{( \frac{x - y}{x} ) ^{2} + 1} \cdot \frac{- x \frac{d y}{d x} + y}{2 x ^{\frac{3}{2}} x - y}

簡素化

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

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