derivative f(x)=(x^3+2)(3x^2+1)

解答

导数

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

15 x^{4} + 3 x^{2} + 12 x

求解步骤

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

使用微分乘法定则:

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

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

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

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

\frac{d}{d x} (3 x^{2} + 1) = 6 x

= 3 x^{2} (3 x^{2} + 1) + 6 x (x^{3} + 2)

化简

3 x^{2} (3 x^{2} + 1) + 6 x (x^{3} + 2) : 15 x^{4} + 3 x^{2} + 12 x

= 15 x^{4} + 3 x^{2} + 12 x

\int \frac{x ^{3}}{( 3 - x ^{4} ) ^{2}} d x

d er i v a t i v e y = 2 sin (4 x)

y^{^{'}} - \frac{8}{x} y = \frac{y ^{3}}{x ^{11}}

x \to 3 - lim (∣ x - 3 ∣)

\frac{d y}{d t} = 3 y (y - 2)