derivative of (3x^2+2y^2/((x-2y)^2))

解

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

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

解答ステップ

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

商の法則を適用:

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

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

\frac{d}{d x} (3 x^{2} + 2 y^{2}) = 6 x + 4 y \frac{d y}{d x}

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

= \frac{( 6 x + 4 y \frac{d y}{d x} ) ( x - 2 y ) ^{2} - 2 ( x - 2 y ) ( 1 - 2 \frac{d y}{d x} ) ( 3 x ^{2} + 2 y ^{2} )}{( ( x - 2 y ) ^{2} ) ^{2}}

簡素化

\frac{( 6 x + 4 y \frac{d y}{d x} ) ( x - 2 y ) ^{2} - 2 ( x - 2 y ) ( 1 - 2 \frac{d y}{d x} ) ( 3 x ^{2} + 2 y ^{2} )}{( ( x - 2 y ) ^{2} ) ^{2}} : \frac{4 ( x \frac{d y}{d x} - y ) ( 3 x + y )}{( x - 2 y ) ^{3}}

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