derivative \sqrt[4]{(2x+x^5)^7}

解

導関数

4 (2 x + x^{5})^{7}

\frac{7 ( 2 x + x ^{5} ) ^{6} ( 2 + 5 x ^{4} )}{4 ( ( 2 x + x ^{5} ) ^{7} ) ^{\frac{3}{4}}}

解答ステップ

\frac{d}{d x} (4 (2 x + x^{5})^{7})

連鎖法則を適用:

\frac{1}{4 ( ( 2 x + x ^{5} ) ^{7} ) ^{\frac{3}{4}}} \frac{d}{d x} ((2 x + x^{5})^{7})

= \frac{1}{4 ( ( 2 x + x ^{5} ) ^{7} ) ^{\frac{3}{4}}} \frac{d}{d x} ((2 x + x^{5})^{7})

\frac{d}{d x} ((2 x + x^{5})^{7}) = 7 (2 x + x^{5})^{6} (2 + 5 x^{4})

= \frac{1}{4 ( ( 2 x + x ^{5} ) ^{7} ) ^{\frac{3}{4}}} \cdot 7 (2 x + x^{5})^{6} (2 + 5 x^{4})

簡素化

\frac{1}{4 ( ( 2 x + x ^{5} ) ^{7} ) ^{\frac{3}{4}}} \cdot 7 (2 x + x^{5})^{6} (2 + 5 x^{4}) : \frac{7 ( 2 x + x ^{5} ) ^{6} ( 2 + 5 x ^{4} )}{4 ( ( 2 x + x ^{5} ) ^{7} ) ^{\frac{3}{4}}}

= \frac{7 ( 2 x + x ^{5} ) ^{6} ( 2 + 5 x ^{4} )}{4 ( ( 2 x + x ^{5} ) ^{7} ) ^{\frac{3}{4}}}