derivative of (1/(x^2-5/(x^4))(x+3x^3))

解

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

\frac{14}{x ^{2}} + \frac{15}{x ^{4}} + 3

解答ステップ

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

積の法則を適用:

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

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

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

\frac{d}{d x} (\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) = - \frac{2}{x ^{3}} + \frac{20}{x ^{5}}

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

= (- \frac{2}{x ^{3}} + \frac{20}{x ^{5}}) (x + 3 x^{3}) + (1 + 9 x^{2}) (\frac{1}{x ^{2}} - \frac{5}{x ^{4}})

簡素化

(- \frac{2}{x ^{3}} + \frac{20}{x ^{5}}) (x + 3 x^{3}) + (1 + 9 x^{2}) (\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) : \frac{14}{x ^{2}} + \frac{15}{x ^{4}} + 3

= \frac{14}{x ^{2}} + \frac{15}{x ^{4}} + 3