derivative of (x^{3/2}-4x(x^4-3/(x^2)+2))

解

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

\frac{11 x ^{10} + 6 x ^{6} + 3 x ^{4} - 40 x ^{\frac{19}{2}} - 24 x ^{\frac{7}{2}}}{2 x ^{\frac{11}{2}}} - 8

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

(\frac{3 x ^{\frac{1}{2}}}{2} - 4) (x^{4} - \frac{3}{x ^{2}} + 2) + (4 x^{3} + \frac{6}{x ^{3}}) (x^{\frac{3}{2}} - 4 x) : \frac{11 x ^{10} + 6 x ^{6} + 3 x ^{4} - 40 x ^{\frac{19}{2}} - 24 x ^{\frac{7}{2}}}{2 x ^{\frac{11}{2}}} - 8

= \frac{11 x ^{10} + 6 x ^{6} + 3 x ^{4} - 40 x ^{\frac{19}{2}} - 24 x ^{\frac{7}{2}}}{2 x ^{\frac{11}{2}}} - 8