derivative of ((3x^7-7x^2+113)/((5x^3)))

解

\frac{d}{d x} (\frac{( 3 x ^{7} - 7 x ^{2} + 113 )}{( 5 x ^{3} )})

\frac{12 x ^{7} + 7 x ^{2} - 339}{5 x ^{4}}

解答ステップ

\frac{d}{d x} (\frac{( 3 x ^{7} - 7 x ^{2} + 113 )}{( 5 x ^{3} )})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{5} \frac{d}{d x} (\frac{3 x ^{7} - 7 x ^{2} + 113}{x ^{3}})

商の法則を適用:

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

= \frac{1}{5} \cdot \frac{\frac{d}{d x} ( 3 x ^{7} - 7 x ^{2} + 113 ) x ^{3} - \frac{d}{d x} ( x ^{3} ) ( 3 x ^{7} - 7 x ^{2} + 113 )}{( x ^{3} ) ^{2}}

\frac{d}{d x} (3 x^{7} - 7 x^{2} + 113) = 21 x^{6} - 14 x

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

= \frac{1}{5} \cdot \frac{( 21 x ^{6} - 14 x ) x ^{3} - 3 x ^{2} ( 3 x ^{7} - 7 x ^{2} + 113 )}{( x ^{3} ) ^{2}}

簡素化

\frac{1}{5} \cdot \frac{( 21 x ^{6} - 14 x ) x ^{3} - 3 x ^{2} ( 3 x ^{7} - 7 x ^{2} + 113 )}{( x ^{3} ) ^{2}} : \frac{12 x ^{7} + 7 x ^{2} - 339}{5 x ^{4}}

= \frac{12 x ^{7} + 7 x ^{2} - 339}{5 x ^{4}}