derivative f(x)=(4x-2x^3)(4+sqrt(x))

解

導関数

f (x) = (4 x - 2 x^{3}) (4 + x)

- 24 x^{2} - 7 x^{2} x + 6 x + 16

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} (4 + x) = \frac{1}{2 x}

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

簡素化

(4 - 6 x^{2}) (4 + x) + \frac{1}{2 x} (4 x - 2 x^{3}) : - 24 x^{2} - 7 x^{2} x + 6 x + 16

= - 24 x^{2} - 7 x^{2} x + 6 x + 16