(\partial)/(\partial x)((4x-2y)/(2x^2+3y))

解

\frac{\partial}{\partial x} (\frac{4 x - 2 y}{2 x ^{2} + 3 y})

\frac{- 8 x ^{2} + 8 x y + 12 y}{( 2 x ^{2} + 3 y ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{4 x - 2 y}{2 x ^{2} + 3 y})

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 - 2 y ) ( 2 x ^{2} + 3 y ) - \frac{\partial}{\partial x} ( 2 x ^{2} + 3 y ) ( 4 x - 2 y )}{( 2 x ^{2} + 3 y ) ^{2}}

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

\frac{\partial}{\partial x} (2 x^{2} + 3 y) = 4 x

= \frac{4 ( 2 x ^{2} + 3 y ) - 4 x ( 4 x - 2 y )}{( 2 x ^{2} + 3 y ) ^{2}}

拡張

4 (2 x^{2} + 3 y) - 4 x (4 x - 2 y) : - 8 x^{2} + 8 x y + 12 y

= \frac{- 8 x ^{2} + 8 x y + 12 y}{( 2 x ^{2} + 3 y ) ^{2}}