(\partial)/(\partial x)((-3x-y)/(y^2+2x^2))

解

\frac{\partial}{\partial x} (\frac{- 3 x - y}{y ^{2} + 2 x ^{2}})

\frac{- 3 y ^{2} + 6 x ^{2} + 4 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{- 3 x - y}{y ^{2} + 2 x ^{2}})

y

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{\partial}{\partial x} ( - 3 x - y ) ( y ^{2} + 2 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 2 x ^{2} ) ( - 3 x - y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (- 3 x - y) = - 3

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

= \frac{( - 3 ) ( y ^{2} + 2 x ^{2} ) - 4 x ( - 3 x - y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

簡素化

\frac{( - 3 ) ( y ^{2} + 2 x ^{2} ) - 4 x ( - 3 x - y )}{( y ^{2} + 2 x ^{2} ) ^{2}} : \frac{- 3 y ^{2} + 6 x ^{2} + 4 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}

= \frac{- 3 y ^{2} + 6 x ^{2} + 4 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}