derivative of (x^3-x(2x-x^2))

解答

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

- 5 x^{4} + 8 x^{3} + 3 x^{2} - 4 x

求解步骤

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

使用微分乘法定则:

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

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

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

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

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

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

化简

(3 x^{2} - 1) (2 x - x^{2}) + (2 - 2 x) (x^{3} - x) : - 5 x^{4} + 8 x^{3} + 3 x^{2} - 4 x

= - 5 x^{4} + 8 x^{3} + 3 x^{2} - 4 x