derivative of ((x^2-2x^3)/(5x))

解

\frac{d}{d x} (\frac{( x ^{2} - 2 x ) ^{3}}{5 x})

\frac{x ( x - 2 ) ^{2} ( 5 x - 4 )}{5}

解答ステップ

\frac{d}{d x} (\frac{( x ^{2} - 2 x ) ^{3}}{5 x})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{5} \frac{d}{d x} (\frac{( x ^{2} - 2 x ) ^{3}}{x})

商の法則を適用:

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

= \frac{1}{5} \cdot \frac{\frac{d}{d x} ( ( x ^{2} - 2 x ) ^{3} ) x - \frac{d x}{d x} ( x ^{2} - 2 x ) ^{3}}{x ^{2}}

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

\frac{d x}{d x} = 1

= \frac{1}{5} \cdot \frac{3 ( x ^{2} - 2 x ) ^{2} ( 2 x - 2 ) x - 1 \cdot ( x ^{2} - 2 x ) ^{3}}{x ^{2}}

簡素化

\frac{1}{5} \cdot \frac{3 ( x ^{2} - 2 x ) ^{2} ( 2 x - 2 ) x - 1 \cdot ( x ^{2} - 2 x ) ^{3}}{x ^{2}} : \frac{x ( x - 2 ) ^{2} ( 5 x - 4 )}{5}

= \frac{x ( x - 2 ) ^{2} ( 5 x - 4 )}{5}