derivative of (3sqrt(x)+x(2-x^2))

解

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

- 3 x^{2} - 6 x x - \frac{3 x ^{\frac{3}{2}}}{2} + \frac{3}{x} + 2

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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