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

解

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

\frac{6 ( x ^{4} + 5 )}{x ^{4}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

\frac{d}{d x} (3 x^{4} - 5) = 12 x^{3}

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

= 2 \cdot \frac{12 x ^{3} x ^{3} - 3 x ^{2} ( 3 x ^{4} - 5 )}{( x ^{3} ) ^{2}}

簡素化

2 \cdot \frac{12 x ^{3} x ^{3} - 3 x ^{2} ( 3 x ^{4} - 5 )}{( x ^{3} ) ^{2}} : \frac{6 ( x ^{4} + 5 )}{x ^{4}}

= \frac{6 ( x ^{4} + 5 )}{x ^{4}}