derivative of (-2x-y/(x+2y))

解

\frac{d}{d x} (\frac{- 2 x - y}{x + 2 y})

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

解答ステップ

\frac{d}{d x} (\frac{- 2 x - y}{x + 2 y})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{d}{d x} (- 2 x - y) = - 2 - \frac{d y}{d x}

\frac{d}{d x} (x + 2 y) = 1 + 2 \frac{d y}{d x}

= \frac{( - 2 - \frac{d y}{d x} ) ( x + 2 y ) - ( 1 + 2 \frac{d y}{d x} ) ( - 2 x - y )}{( x + 2 y ) ^{2}}

拡張

(- 2 - \frac{d y}{d x}) (x + 2 y) - (1 + 2 \frac{d y}{d x}) (- 2 x - y) : 3 x \frac{d y}{d x} - 3 y

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