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

解答

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

\frac{- 8 x ^{\frac{5}{3}} + 10 x ^{\frac{2}{3}}}{3}

求解步骤

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

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{\frac{2}{3}}, g = 2 x - x^{2}

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

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

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

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

化简

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

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

\frac{\partial ^{3}}{\partial x \partial y \partial z} (e^{x y z^{6}})

\frac{\partial}{\partial y} (3 e^{x} ln (y))

\int_{0}^{1} 6 x^{4} e^{- x} d x

x \to 0 lim (\frac{e ^{- 9 x} - 1}{x})

x \to 3 lim (5)