(\partial)/(\partial x)(-x/(2x^2+y^2))

解

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

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

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

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

\frac{\partial}{\partial x} (x) = 1

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

= - \frac{1 \cdot ( 2 x ^{2} + y ^{2} ) - 4 xx}{( 2 x ^{2} + y ^{2} ) ^{2}}

1 \cdot (2 x^{2} + y^{2}) - 4 xx = - 2 x^{2} + y^{2}

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