derivative of sqrt(x^3+1)*\sqrt[3]{2x^5}

解

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

\frac{9 x ^{3} 3 x ^{2} + 10 x ^{\frac{2}{3}} ( x ^{3} + 1 )}{3 \cdot 2 ^{\frac{2}{3}} x ^{3} + 1}

解答ステップ

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

積の法則を適用:

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

f = x^{3} + 1, g = 3 2 x^{5}

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

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

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

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

簡素化

\frac{3 x ^{2}}{2 x ^{3} + 1} 3 2 x^{5} + \frac{5 \cdot 2 ^{\frac{1}{3}} x ^{\frac{2}{3}}}{3} x^{3} + 1 : \frac{9 x ^{3} 3 x ^{2} + 10 x ^{\frac{2}{3}} ( x ^{3} + 1 )}{3 \cdot 2 ^{\frac{2}{3}} x ^{3} + 1}

= \frac{9 x ^{3} 3 x ^{2} + 10 x ^{\frac{2}{3}} ( x ^{3} + 1 )}{3 \cdot 2 ^{\frac{2}{3}} x ^{3} + 1}