derivative (2x^3-6)/(x^4)

解

導関数

\frac{2 x ^{3} - 6}{x ^{4}}

\frac{2 ( - x ^{3} + 12 )}{x ^{5}}

解答ステップ

\frac{d}{d x} (\frac{2 x ^{3} - 6}{x ^{4}})

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 ^{3} - 6 ) x ^{4} - \frac{d}{d x} ( x ^{4} ) ( 2 x ^{3} - 6 )}{( x ^{4} ) ^{2}}

\frac{d}{d x} (2 x^{3} - 6) = 6 x^{2}

\frac{d}{d x} (x^{4}) = 4 x^{3}

= \frac{6 x ^{2} x ^{4} - 4 x ^{3} ( 2 x ^{3} - 6 )}{( x ^{4} ) ^{2}}

簡素化

\frac{6 x ^{2} x ^{4} - 4 x ^{3} ( 2 x ^{3} - 6 )}{( x ^{4} ) ^{2}} : \frac{2 ( - x ^{3} + 12 )}{x ^{5}}

= \frac{2 ( - x ^{3} + 12 )}{x ^{5}}