derivative of x(5x+1^{1/3})

解

\frac{d}{d x} (x (5 x + 1)^{\frac{1}{3}})

\frac{20 x + 3}{3 ( 5 x + 1 ) ^{\frac{2}{3}}}

解答ステップ

\frac{d}{d x} (x (5 x + 1)^{\frac{1}{3}})

積の法則を適用:

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

f = x, g = (5 x + 1)^{\frac{1}{3}}

= \frac{d x}{d x} (5 x + 1)^{\frac{1}{3}} + \frac{d}{d x} ((5 x + 1)^{\frac{1}{3}}) x

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

\frac{d}{d x} ((5 x + 1)^{\frac{1}{3}}) = \frac{5}{3 ( 5 x + 1 ) ^{\frac{2}{3}}}

= 1 \cdot (5 x + 1)^{\frac{1}{3}} + \frac{5}{3 ( 5 x + 1 ) ^{\frac{2}{3}}} x

簡素化

1 \cdot (5 x + 1)^{\frac{1}{3}} + \frac{5}{3 ( 5 x + 1 ) ^{\frac{2}{3}}} x : \frac{20 x + 3}{3 ( 5 x + 1 ) ^{\frac{2}{3}}}

= \frac{20 x + 3}{3 ( 5 x + 1 ) ^{\frac{2}{3}}}