derivative f(x)=(7x^3+x^2)^{2/3}

解

導関数

f (x) = (7 x^{3} + x^{2})^{\frac{2}{3}}

\frac{2 ( 21 x ^{2} + 2 x )}{3 ( 7 x ^{3} + x ^{2} ) ^{\frac{1}{3}}}

解答ステップ

\frac{d}{d x} ((7 x^{3} + x^{2})^{\frac{2}{3}})

連鎖法則を適用:

\frac{2}{3 ( 7 x ^{3} + x ^{2} ) ^{\frac{1}{3}}} \frac{d}{d x} (7 x^{3} + x^{2})

= \frac{2}{3 ( 7 x ^{3} + x ^{2} ) ^{\frac{1}{3}}} \frac{d}{d x} (7 x^{3} + x^{2})

\frac{d}{d x} (7 x^{3} + x^{2}) = 21 x^{2} + 2 x

= \frac{2}{3 ( 7 x ^{3} + x ^{2} ) ^{\frac{1}{3}}} (21 x^{2} + 2 x)

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 ( 21 x ^{2} + 2 x )}{3 ( 7 x ^{3} + x ^{2} ) ^{\frac{1}{3}}}