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

解

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

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

解答ステップ

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

商の法則を適用:

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

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

\frac{d x}{d x} = 1

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

= \frac{1 \cdot ( y ^{2} + x ^{2} ) - ( 2 y \frac{d y}{d x} + 2 x ) x}{( y ^{2} + x ^{2} ) ^{2}}

簡素化

\frac{1 \cdot ( y ^{2} + x ^{2} ) - ( 2 y \frac{d y}{d x} + 2 x ) x}{( y ^{2} + x ^{2} ) ^{2}} : \frac{- x ^{2} - 2 x y \frac{d y}{d x} + y ^{2}}{( y ^{2} + x ^{2} ) ^{2}}

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