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

解

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

\frac{- 12 x ^{2} + 16 x y + 3 y}{( 4 x ^{2} + y ) ^{2}}

解答ステップ

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

\frac{\partial}{\partial x} (3 x - 2 y) = 3

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

= \frac{3 ( 4 x ^{2} + y ) - 8 x ( 3 x - 2 y )}{( 4 x ^{2} + y ) ^{2}}

拡張

3 (4 x^{2} + y) - 8 x (3 x - 2 y) : - 12 x^{2} + 16 x y + 3 y

= \frac{- 12 x ^{2} + 16 x y + 3 y}{( 4 x ^{2} + y ) ^{2}}