derivative f(x)=\sqrt[3]{10x^4+11x^3}

解

導関数

f (x) = 3 10 x^{4} + 11 x^{3}

\frac{40 x ^{3} + 33 x ^{2}}{3 ( x ^{3} ) ^{\frac{2}{3}} ( 10 x + 11 ) ^{\frac{2}{3}}}

解答ステップ

\frac{d}{d x} (3 10 x^{4} + 11 x^{3})

連鎖法則を適用:

\frac{1}{3 ( 10 x ^{4} + 11 x ^{3} ) ^{\frac{2}{3}}} \frac{d}{d x} (10 x^{4} + 11 x^{3})

= \frac{1}{3 ( 10 x ^{4} + 11 x ^{3} ) ^{\frac{2}{3}}} \frac{d}{d x} (10 x^{4} + 11 x^{3})

\frac{d}{d x} (10 x^{4} + 11 x^{3}) = 40 x^{3} + 33 x^{2}

= \frac{1}{3 ( 10 x ^{4} + 11 x ^{3} ) ^{\frac{2}{3}}} (40 x^{3} + 33 x^{2})

簡素化

\frac{1}{3 ( 10 x ^{4} + 11 x ^{3} ) ^{\frac{2}{3}}} (40 x^{3} + 33 x^{2}) : \frac{40 x ^{3} + 33 x ^{2}}{3 ( x ^{3} ) ^{\frac{2}{3}} ( 10 x + 11 ) ^{\frac{2}{3}}}

= \frac{40 x ^{3} + 33 x ^{2}}{3 ( x ^{3} ) ^{\frac{2}{3}} ( 10 x + 11 ) ^{\frac{2}{3}}}