derivative of y/(y+x^2)

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

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

拡張

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

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