derivative of 2\sqrt[5]{(x^3+1^2})

解

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

\frac{12 x ^{2}}{5 ( x ^{3} + 1 ) ^{\frac{3}{5}}}

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

連鎖法則を適用:

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

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

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

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

簡素化

2 \cdot \frac{1}{5 ( ( x ^{3} + 1 ) ^{2} ) ^{\frac{4}{5}}} \cdot 6 x^{2} (x^{3} + 1) : \frac{12 x ^{2}}{5 ( x ^{3} + 1 ) ^{\frac{3}{5}}}

= \frac{12 x ^{2}}{5 ( x ^{3} + 1 ) ^{\frac{3}{5}}}