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

解

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

\frac{- 15 x - x ^{\frac{1}{2}} + 3 x ^{\frac{1}{2}} x + 20 x ^{\frac{3}{2}} - 2 x}{2}

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

= \frac{- 15 x - x ^{\frac{1}{2}} + 3 x ^{\frac{1}{2}} x + 20 x ^{\frac{3}{2}} - 2 x}{2}