derivative of \sqrt[3]{x+sqrt(x)}

解

\frac{d}{d x} (3 x + x)

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

解答ステップ

\frac{d}{d x} (3 x + x)

連鎖法則を適用:

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

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

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

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

簡素化

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

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