derivative f(x)=((4x^5+8))/(x^3)

解

導関数

f (x) = \frac{( 4 x ^{5} + 8 )}{x ^{3}}

\frac{8 ( x ^{5} - 3 )}{x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{4 x ^{5} + 8}{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} ( 4 x ^{5} + 8 ) x ^{3} - \frac{d}{d x} ( x ^{3} ) ( 4 x ^{5} + 8 )}{( x ^{3} ) ^{2}}

\frac{d}{d x} (4 x^{5} + 8) = 20 x^{4}

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

= \frac{20 x ^{4} x ^{3} - 3 x ^{2} ( 4 x ^{5} + 8 )}{( x ^{3} ) ^{2}}

簡素化

\frac{20 x ^{4} x ^{3} - 3 x ^{2} ( 4 x ^{5} + 8 )}{( x ^{3} ) ^{2}} : \frac{8 ( x ^{5} - 3 )}{x ^{4}}

= \frac{8 ( x ^{5} - 3 )}{x ^{4}}