derivative of (1/(x^2-6/(x^4))(x+8x^3))

解答

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{6}{x ^{4}}) (x + 8 x^{3}))

\frac{47}{x ^{2}} + \frac{18}{x ^{4}} + 8

求解步骤

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{6}{x ^{4}}) (x + 8 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{6}{x ^{4}}, g = x + 8 x^{3}

= \frac{d}{d x} (\frac{1}{x ^{2}} - \frac{6}{x ^{4}}) (x + 8 x^{3}) + \frac{d}{d x} (x + 8 x^{3}) (\frac{1}{x ^{2}} - \frac{6}{x ^{4}})

\frac{d}{d x} (\frac{1}{x ^{2}} - \frac{6}{x ^{4}}) = - \frac{2}{x ^{3}} + \frac{24}{x ^{5}}

\frac{d}{d x} (x + 8 x^{3}) = 1 + 24 x^{2}

= (- \frac{2}{x ^{3}} + \frac{24}{x ^{5}}) (x + 8 x^{3}) + (1 + 24 x^{2}) (\frac{1}{x ^{2}} - \frac{6}{x ^{4}})

化简

(- \frac{2}{x ^{3}} + \frac{24}{x ^{5}}) (x + 8 x^{3}) + (1 + 24 x^{2}) (\frac{1}{x ^{2}} - \frac{6}{x ^{4}}) : \frac{47}{x ^{2}} + \frac{18}{x ^{4}} + 8

= \frac{47}{x ^{2}} + \frac{18}{x ^{4}} + 8

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

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