derivative f(x)=(x^3+9)/(x^3)

解

導関数

f (x) = \frac{x ^{3} + 9}{x ^{3}}

- \frac{27}{x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{x ^{3} + 9}{x ^{3}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( x ^{3} + 9 ) x ^{3} - \frac{d}{d x} ( x ^{3} ) ( x ^{3} + 9 )}{( x ^{3} ) ^{2}}

\frac{d}{d x} (x^{3} + 9) = 3 x^{2}

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

= \frac{3 x ^{2} x ^{3} - 3 x ^{2} ( x ^{3} + 9 )}{( x ^{3} ) ^{2}}

簡素化

\frac{3 x ^{2} x ^{3} - 3 x ^{2} ( x ^{3} + 9 )}{( x ^{3} ) ^{2}} : - \frac{27}{x ^{4}}

= - \frac{27}{x ^{4}}