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

解

導関数

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

\frac{15 x ^{2} ( x + 5 )}{( 5 - 3 x ) ^{5}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

5 \cdot \frac{3 x ^{2} ( 5 - 3 x ) ^{4} - ( - 12 ( 5 - 3 x ) ^{3} ) x ^{3}}{( ( 5 - 3 x ) ^{4} ) ^{2}} : \frac{15 x ^{2} ( x + 5 )}{( 5 - 3 x ) ^{5}}

= \frac{15 x ^{2} ( x + 5 )}{( 5 - 3 x ) ^{5}}