derivative of 4x\sqrt[3]{x^4}

解

\frac{d}{d x} (4 x 3 x^{4})

4 (x 3 x + \frac{4 x ^{\frac{4}{3}}}{3})

解答ステップ

\frac{d}{d x} (4 x 3 x^{4})

定数を除去:

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

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

積の法則を適用:

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

f = x, g = 3 x^{4}

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

\frac{d x}{d x} = 1

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

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

簡素化

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

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