derivative of-x/(x^2+y^2)

解

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

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

解答ステップ

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

定数を除去:

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

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

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

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

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

簡素化

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

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