derivative of 1/2 x^2\sqrt[3]{x}

解

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

x 3 x + \frac{x ^{\frac{4}{3}}}{6}

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

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

f = x^{2}, g = 3 x

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

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

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

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

簡素化

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

= x 3 x + \frac{x ^{\frac{4}{3}}}{6}