derivative f(x)=(x^7+5x)^{1/3}

解

導関数

f (x) = (x^{7} + 5 x)^{\frac{1}{3}}

\frac{7 x ^{6} + 5}{3 ( x ^{7} + 5 x ) ^{\frac{2}{3}}}

解答ステップ

\frac{d}{d x} ((x^{7} + 5 x)^{\frac{1}{3}})

連鎖法則を適用:

\frac{1}{3 ( x ^{7} + 5 x ) ^{\frac{2}{3}}} \frac{d}{d x} (x^{7} + 5 x)

= \frac{1}{3 ( x ^{7} + 5 x ) ^{\frac{2}{3}}} \frac{d}{d x} (x^{7} + 5 x)

\frac{d}{d x} (x^{7} + 5 x) = 7 x^{6} + 5

= \frac{1}{3 ( x ^{7} + 5 x ) ^{\frac{2}{3}}} (7 x^{6} + 5)

簡素化

\frac{1}{3 ( x ^{7} + 5 x ) ^{\frac{2}{3}}} (7 x^{6} + 5) : \frac{7 x ^{6} + 5}{3 ( x ^{7} + 5 x ) ^{\frac{2}{3}}}

= \frac{7 x ^{6} + 5}{3 ( x ^{7} + 5 x ) ^{\frac{2}{3}}}