derivative of (x+1^{2/3}(3x^2-2)^3)

解

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

\frac{2 ( 3 x ^{2} - 2 ) ^{3} + 54 x ( 3 x ^{2} - 2 ) ^{2} ( x + 1 )}{3 ( x + 1 ) ^{\frac{1}{3}}}

解答ステップ

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

積の法則を適用:

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

f = (x + 1)^{\frac{2}{3}}, g = (3 x^{2} - 2)^{3}

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

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

\frac{d}{d x} ((3 x^{2} - 2)^{3}) = 18 x (3 x^{2} - 2)^{2}

= \frac{2}{3 ( x + 1 ) ^{\frac{1}{3}}} (3 x^{2} - 2)^{3} + 18 x (3 x^{2} - 2)^{2} (x + 1)^{\frac{2}{3}}

簡素化

\frac{2}{3 ( x + 1 ) ^{\frac{1}{3}}} (3 x^{2} - 2)^{3} + 18 x (3 x^{2} - 2)^{2} (x + 1)^{\frac{2}{3}} : \frac{2 ( 3 x ^{2} - 2 ) ^{3} + 54 x ( 3 x ^{2} - 2 ) ^{2} ( x + 1 )}{3 ( x + 1 ) ^{\frac{1}{3}}}

= \frac{2 ( 3 x ^{2} - 2 ) ^{3} + 54 x ( 3 x ^{2} - 2 ) ^{2} ( x + 1 )}{3 ( x + 1 ) ^{\frac{1}{3}}}