derivative 12x\sqrt[3]{1-x}sqrt(x+1)

解答

导数

12 x 3 1 - x x + 1

\frac{2 ( - 2 x ^{2} + 9 x 3 - x + 1 ( - x + 1 ) ^{\frac{2}{3}} - 2 x + 6 3 - x + 1 ( - x + 1 ) ^{\frac{2}{3}} )}{( - x + 1 ) ^{\frac{2}{3}} x + 1}

求解步骤

\frac{d}{d x} (12 x 3 1 - x x + 1)

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 12 \frac{d}{d x} (x 3 1 - x x + 1)

使用微分乘法定则:

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

f = x, g = 3 1 - x x + 1

= 12 (\frac{d x}{d x} 3 1 - x x + 1 + \frac{d}{d x} (3 1 - x x + 1) x)

\frac{d x}{d x} = 1

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

= 12 (1 \cdot 3 1 - x x + 1 + (- \frac{x + 1}{3 ( 1 - x ) ^{\frac{2}{3}}} + \frac{3 1 - x}{2 x + 1}) x)

化简

12 (1 \cdot 3 1 - x x + 1 + (- \frac{x + 1}{3 ( 1 - x ) ^{\frac{2}{3}}} + \frac{3 1 - x}{2 x + 1}) x) : \frac{2 ( - 2 x ^{2} + 9 x 3 - x + 1 ( - x + 1 ) ^{\frac{2}{3}} - 2 x + 6 3 - x + 1 ( - x + 1 ) ^{\frac{2}{3}} )}{( - x + 1 ) ^{\frac{2}{3}} x + 1}

= \frac{2 ( - 2 x ^{2} + 9 x 3 - x + 1 ( - x + 1 ) ^{\frac{2}{3}} - 2 x + 6 3 - x + 1 ( - x + 1 ) ^{\frac{2}{3}} )}{( - x + 1 ) ^{\frac{2}{3}} x + 1}

\int_{0}^{l n (3)} e^{x} - e^{- 4 x} d x

\frac{\partial}{\partial y} (1 + y^{2} sin (2 x))

\int x (1 + x^{2}) d x

\frac{d}{d x} (x^{3} - 12 x^{2} + 45 x)

11 x - 4 y x^{2} + 1 y^{^{'}} = 0, y (0) = 2