(\partial)/(\partial x)((5x-9y)/(5x+9y))

解

\frac{\partial}{\partial x} (\frac{5 x - 9 y}{5 x + 9 y})

\frac{90 y}{( 5 x + 9 y ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{5 x - 9 y}{5 x + 9 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} ( 5 x - 9 y ) ( 5 x + 9 y ) - \frac{\partial}{\partial x} ( 5 x + 9 y ) ( 5 x - 9 y )}{( 5 x + 9 y ) ^{2}}

\frac{\partial}{\partial x} (5 x - 9 y) = 5

\frac{\partial}{\partial x} (5 x + 9 y) = 5

= \frac{5 ( 5 x + 9 y ) - 5 ( 5 x - 9 y )}{( 5 x + 9 y ) ^{2}}

拡張

5 (5 x + 9 y) - 5 (5 x - 9 y) : 90 y

= \frac{90 y}{( 5 x + 9 y ) ^{2}}