(\partial)/(\partial y)((-x-7y)/(y^2+2x^2))

解

\frac{\partial}{\partial y} (\frac{- x - 7 y}{y ^{2} + 2 x ^{2}})

\frac{7 y ^{2} + 2 x y - 14 x ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial y} (\frac{- x - 7 y}{y ^{2} + 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{\partial}{\partial y} ( - x - 7 y ) ( y ^{2} + 2 x ^{2} ) - \frac{\partial}{\partial y} ( y ^{2} + 2 x ^{2} ) ( - x - 7 y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{\partial}{\partial y} (- x - 7 y) = - 7

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

= \frac{( - 7 ) ( y ^{2} + 2 x ^{2} ) - 2 y ( - x - 7 y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

簡素化

\frac{( - 7 ) ( y ^{2} + 2 x ^{2} ) - 2 y ( - x - 7 y )}{( y ^{2} + 2 x ^{2} ) ^{2}} : \frac{7 y ^{2} + 2 x y - 14 x ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

= \frac{7 y ^{2} + 2 x y - 14 x ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}