derivative f(x)=(x^3+3x+6)(4+1/(x^2))

解答

导数

f (x) = (x^{3} + 3 x + 6) (4 + \frac{1}{x ^{2}})

\frac{12 x ^{5} - 2 x ^{3} - 3 x - 12}{x ^{3}} + 15

求解步骤

\frac{d}{d x} ((x^{3} + 3 x + 6) (4 + \frac{1}{x ^{2}}))

使用微分乘法定则:

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

f = x^{3} + 3 x + 6, g = 4 + \frac{1}{x ^{2}}

= \frac{d}{d x} (x^{3} + 3 x + 6) (4 + \frac{1}{x ^{2}}) + \frac{d}{d x} (4 + \frac{1}{x ^{2}}) (x^{3} + 3 x + 6)

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

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

= (3 x^{2} + 3) (4 + \frac{1}{x ^{2}}) + (- \frac{2}{x ^{3}}) (x^{3} + 3 x + 6)

化简

(3 x^{2} + 3) (4 + \frac{1}{x ^{2}}) + (- \frac{2}{x ^{3}}) (x^{3} + 3 x + 6) : \frac{12 x ^{5} - 2 x ^{3} - 3 x - 12}{x ^{3}} + 15

= \frac{12 x ^{5} - 2 x ^{3} - 3 x - 12}{x ^{3}} + 15

\int cos (\frac{x}{4}) sin (\frac{x}{4}) d x

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