derivative (480x)/((x^2+5)^3)

解

導関数

\frac{480 x}{( x ^{2} + 5 ) ^{3}}

\frac{2400 ( - x ^{2} + 1 )}{( x ^{2} + 5 ) ^{4}}

解答ステップ

\frac{d}{d x} (\frac{480 x}{( x ^{2} + 5 ) ^{3}})

定数を除去:

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

= 480 \frac{d}{d x} (\frac{x}{( x ^{2} + 5 ) ^{3}})

商の法則を適用:

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

= 480 \cdot \frac{\frac{d x}{d x} ( x ^{2} + 5 ) ^{3} - \frac{d}{d x} ( ( x ^{2} + 5 ) ^{3} ) x}{( ( x ^{2} + 5 ) ^{3} ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} ((x^{2} + 5)^{3}) = 6 x (x^{2} + 5)^{2}

= 480 \cdot \frac{1 \cdot ( x ^{2} + 5 ) ^{3} - 6 x ( x ^{2} + 5 ) ^{2} x}{( ( x ^{2} + 5 ) ^{3} ) ^{2}}

簡素化

480 \cdot \frac{1 \cdot ( x ^{2} + 5 ) ^{3} - 6 x ( x ^{2} + 5 ) ^{2} x}{( ( x ^{2} + 5 ) ^{3} ) ^{2}} : \frac{2400 ( - x ^{2} + 1 )}{( x ^{2} + 5 ) ^{4}}

= \frac{2400 ( - x ^{2} + 1 )}{( x ^{2} + 5 ) ^{4}}