d/(dy)((x^2-y^2)/(x^2+y^2))

解

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

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

解答ステップ

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

x

を定数として扱う

商の法則を適用:

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

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

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

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

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

簡素化

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

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