derivative of (4x^{3/2}-2x^{1/2}^3)

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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