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

解答

导数

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}}}

求解步骤

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

化简

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})

使用微分加减法定则:

(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}

合并分式

\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}}}

\int - 7 (- 7 x)^{3} d x

\frac{d}{d x} (\frac{6}{7 x ^{4}})

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

\frac{\partial}{\partial y} (\frac{2 x - 2 z}{3 y + 2 z})

d er i v a t i v e y = 2 x^{2} - 4 x + 2