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

解

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

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

解答ステップ

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

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

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

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

拡張

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

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