derivative y=(3x^4+7)/(x^2)

解

導関数

y = \frac{3 x ^{4} + 7}{x ^{2}}

\frac{2 ( 3 x ^{4} - 7 )}{x ^{3}}

解答ステップ

\frac{d}{d x} (\frac{3 x ^{4} + 7}{x ^{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 ^{4} + 7 ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( 3 x ^{4} + 7 )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (3 x^{4} + 7) = 12 x^{3}

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

= \frac{12 x ^{3} x ^{2} - 2 x ( 3 x ^{4} + 7 )}{( x ^{2} ) ^{2}}

簡素化

\frac{12 x ^{3} x ^{2} - 2 x ( 3 x ^{4} + 7 )}{( x ^{2} ) ^{2}} : \frac{2 ( 3 x ^{4} - 7 )}{x ^{3}}

= \frac{2 ( 3 x ^{4} - 7 )}{x ^{3}}