derivative of x^2(1-x^3)

解答

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

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

求解步骤

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

使用微分乘法定则:

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

f = x^{2}, g = (1 - x)^{3}

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

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

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

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

化简

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