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

解

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

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

解答ステップ

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

簡素化

\frac{x ^{2} + 1}{3 x ^{2} + 1} : 6 x^{2} + 1

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

連鎖法則を適用:

\frac{1}{6 ( x ^{2} + 1 ) ^{\frac{5}{6}}} \frac{d}{d x} (x^{2} + 1)

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

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

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

簡素化

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

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