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

解

導関数

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

\frac{1}{3} + \frac{8 - 9 x ^{\frac{8}{3}}}{12 x ^{\frac{2}{3}}}

解答ステップ

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

簡素化

2 x^{\frac{1}{3}} + \frac{x}{3} - \frac{1}{4} x^{3} : 2 x^{\frac{1}{3}} + \frac{x}{3} - \frac{x ^{3}}{4}

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

和/差の法則を適用:

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

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

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

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

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

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

分数を組み合わせる

\frac{2}{3 x ^{\frac{2}{3}}} - \frac{3 x ^{2}}{4} : \frac{8 - 9 x ^{\frac{8}{3}}}{12 x ^{\frac{2}{3}}}

= \frac{1}{3} + \frac{8 - 9 x ^{\frac{8}{3}}}{12 x ^{\frac{2}{3}}}