derivative f(x)=(6x^2-x+7)^{-5/3}

解

導関数

f (x) = (6 x^{2} - x + 7)^{- \frac{5}{3}}

- \frac{5 ( 12 x - 1 )}{3 ( 6 x ^{2} - x + 7 ) ^{\frac{8}{3}}}

解答ステップ

\frac{d}{d x} ((6 x^{2} - x + 7)^{- \frac{5}{3}})

連鎖法則を適用:

- \frac{5}{3 ( 6 x ^{2} - x + 7 ) ^{\frac{8}{3}}} \frac{d}{d x} (6 x^{2} - x + 7)

= - \frac{5}{3 ( 6 x ^{2} - x + 7 ) ^{\frac{8}{3}}} \frac{d}{d x} (6 x^{2} - x + 7)

\frac{d}{d x} (6 x^{2} - x + 7) = 12 x - 1

= - \frac{5}{3 ( 6 x ^{2} - x + 7 ) ^{\frac{8}{3}}} (12 x - 1)

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{5 ( 12 x - 1 )}{3 ( 6 x ^{2} - x + 7 ) ^{\frac{8}{3}}}