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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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