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

解

導関数

\frac{y ^{2} + x ^{2}}{y ^{2} - x ^{2}}

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

解答ステップ

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

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

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

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

簡素化

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

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