derivative of (x^3+5x^2-x^{-2}+8^{-7/6})

解

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

- \frac{7 x ^{\frac{4}{3}} ( 3 x ^{5} + 10 x ^{4} + 2 )}{6 ( x ^{5} + 5 x ^{4} - 1 + 8 x ^{2} ) ^{\frac{13}{6}}}

解答ステップ

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

連鎖法則を適用:

- \frac{7}{6 ( x ^{3} + 5 x ^{2} - x ^{- 2} + 8 ) ^{\frac{13}{6}}} \frac{d}{d x} (x^{3} + 5 x^{2} - x^{- 2} + 8)

= - \frac{7}{6 ( x ^{3} + 5 x ^{2} - x ^{- 2} + 8 ) ^{\frac{13}{6}}} \frac{d}{d x} (x^{3} + 5 x^{2} - x^{- 2} + 8)

\frac{d}{d x} (x^{3} + 5 x^{2} - x^{- 2} + 8) = 3 x^{2} + 10 x + \frac{2}{x ^{3}}

= - \frac{7}{6 ( x ^{3} + 5 x ^{2} - x ^{- 2} + 8 ) ^{\frac{13}{6}}} (3 x^{2} + 10 x + \frac{2}{x ^{3}})

簡素化

- \frac{7}{6 ( x ^{3} + 5 x ^{2} - x ^{- 2} + 8 ) ^{\frac{13}{6}}} (3 x^{2} + 10 x + \frac{2}{x ^{3}}) : - \frac{7 x ^{\frac{4}{3}} ( 3 x ^{5} + 10 x ^{4} + 2 )}{6 ( x ^{5} + 5 x ^{4} - 1 + 8 x ^{2} ) ^{\frac{13}{6}}}

= - \frac{7 x ^{\frac{4}{3}} ( 3 x ^{5} + 10 x ^{4} + 2 )}{6 ( x ^{5} + 5 x ^{4} - 1 + 8 x ^{2} ) ^{\frac{13}{6}}}