(\partial)/(\partial x)((y-5x)/(y^2+4x^2))

解

\frac{\partial}{\partial x} (\frac{y - 5 x}{y ^{2} + 4 x ^{2}})

\frac{20 x ^{2} - 8 y x - 5 y ^{2}}{( y ^{2} + 4 x ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{y - 5 x}{y ^{2} + 4 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} ( y - 5 x ) ( y ^{2} + 4 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 4 x ^{2} ) ( y - 5 x )}{( y ^{2} + 4 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (y - 5 x) = - 5

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

= \frac{( - 5 ) ( y ^{2} + 4 x ^{2} ) - 8 x ( y - 5 x )}{( y ^{2} + 4 x ^{2} ) ^{2}}

簡素化

\frac{( - 5 ) ( y ^{2} + 4 x ^{2} ) - 8 x ( y - 5 x )}{( y ^{2} + 4 x ^{2} ) ^{2}} : \frac{20 x ^{2} - 8 y x - 5 y ^{2}}{( y ^{2} + 4 x ^{2} ) ^{2}}

= \frac{20 x ^{2} - 8 y x - 5 y ^{2}}{( y ^{2} + 4 x ^{2} ) ^{2}}