(\partial)/(\partial x)((7y-4x)/(y^2+2x^2))

解

\frac{\partial}{\partial x} (\frac{7 y - 4 x}{y ^{2} + 2 x ^{2}})

\frac{8 x ^{2} - 28 y x - 4 y ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{7 y - 4 x}{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} ( 7 y - 4 x ) ( y ^{2} + 2 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 2 x ^{2} ) ( 7 y - 4 x )}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (7 y - 4 x) = - 4

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

= \frac{( - 4 ) ( y ^{2} + 2 x ^{2} ) - 4 x ( 7 y - 4 x )}{( y ^{2} + 2 x ^{2} ) ^{2}}

簡素化

\frac{( - 4 ) ( y ^{2} + 2 x ^{2} ) - 4 x ( 7 y - 4 x )}{( y ^{2} + 2 x ^{2} ) ^{2}} : \frac{8 x ^{2} - 28 y x - 4 y ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

= \frac{8 x ^{2} - 28 y x - 4 y ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}