derivative of (3x^5-7x^2-4/(x^2))

解

\frac{d}{d x} (\frac{3 x ^{5} - 7 x ^{2} - 4}{x ^{2}})

\frac{9 x ^{5} + 8}{x ^{3}}

解答ステップ

\frac{d}{d x} (\frac{3 x ^{5} - 7 x ^{2} - 4}{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 ^{5} - 7 x ^{2} - 4 ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( 3 x ^{5} - 7 x ^{2} - 4 )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (3 x^{5} - 7 x^{2} - 4) = 15 x^{4} - 14 x

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

= \frac{( 15 x ^{4} - 14 x ) x ^{2} - 2 x ( 3 x ^{5} - 7 x ^{2} - 4 )}{( x ^{2} ) ^{2}}

簡素化

\frac{( 15 x ^{4} - 14 x ) x ^{2} - 2 x ( 3 x ^{5} - 7 x ^{2} - 4 )}{( x ^{2} ) ^{2}} : \frac{9 x ^{5} + 8}{x ^{3}}

= \frac{9 x ^{5} + 8}{x ^{3}}