derivative of (1/(x^2-3/(x^4))(x+7x^3))

解答

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{3}{x ^{4}}) (x + 7 x^{3}))

\frac{20}{x ^{2}} + \frac{9}{x ^{4}} + 7

求解步骤

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{3}{x ^{4}}) (x + 7 x^{3}))

使用微分乘法定则:

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

f = \frac{1}{x ^{2}} - \frac{3}{x ^{4}}, g = x + 7 x^{3}

= \frac{d}{d x} (\frac{1}{x ^{2}} - \frac{3}{x ^{4}}) (x + 7 x^{3}) + \frac{d}{d x} (x + 7 x^{3}) (\frac{1}{x ^{2}} - \frac{3}{x ^{4}})

\frac{d}{d x} (\frac{1}{x ^{2}} - \frac{3}{x ^{4}}) = - \frac{2}{x ^{3}} + \frac{12}{x ^{5}}

\frac{d}{d x} (x + 7 x^{3}) = 1 + 21 x^{2}

= (- \frac{2}{x ^{3}} + \frac{12}{x ^{5}}) (x + 7 x^{3}) + (1 + 21 x^{2}) (\frac{1}{x ^{2}} - \frac{3}{x ^{4}})

化简

(- \frac{2}{x ^{3}} + \frac{12}{x ^{5}}) (x + 7 x^{3}) + (1 + 21 x^{2}) (\frac{1}{x ^{2}} - \frac{3}{x ^{4}}) : \frac{20}{x ^{2}} + \frac{9}{x ^{4}} + 7

= \frac{20}{x ^{2}} + \frac{9}{x ^{4}} + 7

\frac{d y}{d x} = x (9 - y)

\frac{d}{d x} (\frac{1 + 3 x}{1 - 3 x})

x \to \infty lim (9)

d er i v a t i v e - 3 cos (x) + 2 sin (x)

\frac{d}{d x} (- 2 sin (x) cos (x))