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

解

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

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

解答ステップ

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

簡素化

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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