(\partial)/(\partial x)((4x-6y)/(y^2+2x^2))

解

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

\frac{4 y ^{2} - 8 x ^{2} + 24 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}

解答ステップ

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

\frac{\partial}{\partial x} (4 x - 6 y) = 4

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

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

拡張

4 (y^{2} + 2 x^{2}) - 4 x (4 x - 6 y) : 4 y^{2} - 8 x^{2} + 24 x y

= \frac{4 y ^{2} - 8 x ^{2} + 24 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}