derivative f(x)=(6x)/((x^3-9)^2)

解

導関数

f (x) = \frac{6 x}{( x ^{3} - 9 ) ^{2}}

\frac{6 ( - 5 x ^{3} - 9 )}{( x ^{3} - 9 ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{6 x}{( x ^{3} - 9 ) ^{2}})

定数を除去:

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

= 6 \frac{d}{d x} (\frac{x}{( x ^{3} - 9 ) ^{2}})

商の法則を適用:

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

= 6 \cdot \frac{\frac{d x}{d x} ( x ^{3} - 9 ) ^{2} - \frac{d}{d x} ( ( x ^{3} - 9 ) ^{2} ) x}{( ( x ^{3} - 9 ) ^{2} ) ^{2}}

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

\frac{d}{d x} ((x^{3} - 9)^{2}) = 6 x^{2} (x^{3} - 9)

= 6 \cdot \frac{1 \cdot ( x ^{3} - 9 ) ^{2} - 6 x ^{2} ( x ^{3} - 9 ) x}{( ( x ^{3} - 9 ) ^{2} ) ^{2}}

簡素化

6 \cdot \frac{1 \cdot ( x ^{3} - 9 ) ^{2} - 6 x ^{2} ( x ^{3} - 9 ) x}{( ( x ^{3} - 9 ) ^{2} ) ^{2}} : \frac{6 ( - 5 x ^{3} - 9 )}{( x ^{3} - 9 ) ^{3}}

= \frac{6 ( - 5 x ^{3} - 9 )}{( x ^{3} - 9 ) ^{3}}