derivative f(x)=(4x-5)(3x^3-x^2+1)

解答

导数

f (x) = (4 x - 5) (3 x^{3} - x^{2} + 1)

48 x^{3} - 57 x^{2} + 10 x + 4

求解步骤

\frac{d}{d x} ((4 x - 5) (3 x^{3} - x^{2} + 1))

使用微分乘法定则:

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

f = 4 x - 5, g = 3 x^{3} - x^{2} + 1

= \frac{d}{d x} (4 x - 5) (3 x^{3} - x^{2} + 1) + \frac{d}{d x} (3 x^{3} - x^{2} + 1) (4 x - 5)

\frac{d}{d x} (4 x - 5) = 4

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

= 4 (3 x^{3} - x^{2} + 1) + (9 x^{2} - 2 x) (4 x - 5)

化简

4 (3 x^{3} - x^{2} + 1) + (9 x^{2} - 2 x) (4 x - 5) : 48 x^{3} - 57 x^{2} + 10 x + 4

= 48 x^{3} - 57 x^{2} + 10 x + 4

\int \frac{x - 3}{x ^{2} - 1} d x

d er i v a t i v e x^{3}

\int \frac{2 x ^{2} + 1}{x} d x

x \to 1 lim (\frac{x ^{4} - 1}{x ^{2}})

d er i v a t i v e f (x) = (2 x^{2} - 4)^{- 1}