derivative (x^3+2x+7)(2+1/(x^2))

解

導関数

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

\frac{6 x ^{5} - 2 x ^{3} - 2 x - 14}{x ^{3}} + 7

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

(3 x^{2} + 2) (2 + \frac{1}{x ^{2}}) + (- \frac{2}{x ^{3}}) (x^{3} + 2 x + 7) : \frac{6 x ^{5} - 2 x ^{3} - 2 x - 14}{x ^{3}} + 7

= \frac{6 x ^{5} - 2 x ^{3} - 2 x - 14}{x ^{3}} + 7