derivative f(x)=(\sqrt[3]{4x^4+5x^3})

解

導関数

f (x) = (3 4 x^{4} + 5 x^{3})

\frac{16 x ^{3} + 15 x ^{2}}{3 ( x ^{3} ) ^{\frac{2}{3}} ( 4 x + 5 ) ^{\frac{2}{3}}}

解答ステップ

\frac{d}{d x} (3 4 x^{4} + 5 x^{3})

連鎖法則を適用:

\frac{1}{3 ( 4 x ^{4} + 5 x ^{3} ) ^{\frac{2}{3}}} \frac{d}{d x} (4 x^{4} + 5 x^{3})

= \frac{1}{3 ( 4 x ^{4} + 5 x ^{3} ) ^{\frac{2}{3}}} \frac{d}{d x} (4 x^{4} + 5 x^{3})

\frac{d}{d x} (4 x^{4} + 5 x^{3}) = 16 x^{3} + 15 x^{2}

= \frac{1}{3 ( 4 x ^{4} + 5 x ^{3} ) ^{\frac{2}{3}}} (16 x^{3} + 15 x^{2})

簡素化

\frac{1}{3 ( 4 x ^{4} + 5 x ^{3} ) ^{\frac{2}{3}}} (16 x^{3} + 15 x^{2}) : \frac{16 x ^{3} + 15 x ^{2}}{3 ( x ^{3} ) ^{\frac{2}{3}} ( 4 x + 5 ) ^{\frac{2}{3}}}

= \frac{16 x ^{3} + 15 x ^{2}}{3 ( x ^{3} ) ^{\frac{2}{3}} ( 4 x + 5 ) ^{\frac{2}{3}}}