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

解

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

\frac{- 3 x ^{2} + 4 x y + 9 y}{( x ^{2} + 3 y ) ^{2}}

解答ステップ

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

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

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

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

拡張

3 (x^{2} + 3 y) - 2 x (3 x - 2 y) : - 3 x^{2} + 4 x y + 9 y

= \frac{- 3 x ^{2} + 4 x y + 9 y}{( x ^{2} + 3 y ) ^{2}}