derivative ((5x^5-3x^2+1))/x

解

導関数

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

\frac{20 x ^{5} - 3 x ^{2} - 1}{x ^{2}}

解答ステップ

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

商の法則を適用:

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

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

\frac{d}{d x} (5 x^{5} - 3 x^{2} + 1) = 25 x^{4} - 6 x

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

= \frac{( 25 x ^{4} - 6 x ) x - 1 \cdot ( 5 x ^{5} - 3 x ^{2} + 1 )}{x ^{2}}

簡素化

\frac{( 25 x ^{4} - 6 x ) x - 1 \cdot ( 5 x ^{5} - 3 x ^{2} + 1 )}{x ^{2}} : \frac{20 x ^{5} - 3 x ^{2} - 1}{x ^{2}}

= \frac{20 x ^{5} - 3 x ^{2} - 1}{x ^{2}}