derivative of 3x^4+2x^{1/3}-1/(x^2)

解

\frac{d}{d x} (3 x^{4} + 2 x^{\frac{1}{3}} - \frac{1}{x ^{2}})

\frac{2 x ^{3} + 6 x ^{\frac{2}{3}} + 36 x ^{\frac{20}{3}}}{3 x ^{\frac{11}{3}}}

解答ステップ

\frac{d}{d x} (3 x^{4} + 2 x^{\frac{1}{3}} - \frac{1}{x ^{2}})

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (2 x^{\frac{1}{3}}) - \frac{d}{d x} (\frac{1}{x ^{2}})

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

\frac{d}{d x} (2 x^{\frac{1}{3}}) = \frac{2}{3 x ^{\frac{2}{3}}}

\frac{d}{d x} (\frac{1}{x ^{2}}) = - \frac{2}{x ^{3}}

= 12 x^{3} + \frac{2}{3 x ^{\frac{2}{3}}} - (- \frac{2}{x ^{3}})

簡素化

12 x^{3} + \frac{2}{3 x ^{\frac{2}{3}}} - (- \frac{2}{x ^{3}}) : \frac{2 x ^{3} + 6 x ^{\frac{2}{3}} + 36 x ^{\frac{20}{3}}}{3 x ^{\frac{11}{3}}}

= \frac{2 x ^{3} + 6 x ^{\frac{2}{3}} + 36 x ^{\frac{20}{3}}}{3 x ^{\frac{11}{3}}}