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

解

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

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

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