derivative f(x)=x(1-x)^4

解答

导数

f (x) = x (1 - x)^{4}

(1 - x)^{4} - 4 x (1 - x)^{3}

求解步骤

\frac{d}{d x} (x (1 - x)^{4})

使用微分乘法定则:

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

f = x, g = (1 - x)^{4}

= \frac{d x}{d x} (1 - x)^{4} + \frac{d}{d x} ((1 - x)^{4}) x

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

\frac{d}{d x} ((1 - x)^{4}) = - 4 (1 - x)^{3}

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

化简

= (1 - x)^{4} - 4 x (1 - x)^{3}

Δ x \to 0 lim (\frac{0}{Δ x ^{2}})

n \to \infty lim (x^{n})

y^{^{'}} + 2 y = 0, y (0) = 0

a re a y = 5 x^{2}, y = x^{2} + 2

x \to 0 lim (x e^{- \frac{1}{x ^{2}}})