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

解

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

\frac{2 y ^{2} - 8 x ^{2} + 16 x y}{( y ^{2} + 4 x ^{2} ) ^{2}}

解答ステップ

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

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

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

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

拡張

2 (y^{2} + 4 x^{2}) - 8 x (2 x - 2 y) : 2 y^{2} - 8 x^{2} + 16 x y

= \frac{2 y ^{2} - 8 x ^{2} + 16 x y}{( y ^{2} + 4 x ^{2} ) ^{2}}