derivative of x(1-x^2)

解答

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

3 x^{2} - 4 x + 1

求解步骤

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

使用微分乘法定则:

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

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

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

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

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

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

化简

1 \cdot (1 - x)^{2} + (- 2 (1 - x)) x : 3 x^{2} - 4 x + 1

= 3 x^{2} - 4 x + 1

(x, y) \to (π, \frac{π}{2}) lim (y sin (x - y))

t an g e n t f (x) = (x^{2} - 8)^{5}, at x = 3

(x, y) \to (0, 0) lim (\frac{y}{x})

t an g e n t f (x) = (x^{2} - 48)^{4}

\frac{d}{d x} (6 x^{\frac{1}{3}})